Theorem biasandor-P4.34a 491

Description: '↔' in Terms of '∧' and '∨'.

Assertion

Ref	Expression
biasandor-P4.34a	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem biasandor-P4.34a

Step	Hyp	Ref	Expression
1		dfbi-P3.47 358	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2		imasor-P4.32a 487	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3		imasor-P4.32a 487	. . . 4 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜓 ∨ 𝜑))
4	2, 3	subandd-P3.42c.RC 344	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜓 ∨ 𝜑)))
5		andoveror-P4.27a 461	. . 3 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜓 ∨ 𝜑)) ↔ (((¬ 𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ∨ ((¬ 𝜑 ∨ 𝜓) ∧ 𝜑)))
6		andcomm-P3.35 314	. . . . 5 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ (¬ 𝜑 ∨ 𝜓)))
7		andoveror-P4.27a 461	. . . . 5 ⊢ ((¬ 𝜓 ∧ (¬ 𝜑 ∨ 𝜓)) ↔ ((¬ 𝜓 ∧ ¬ 𝜑) ∨ (¬ 𝜓 ∧ 𝜓)))
8		andcomm-P3.35 314	. . . . . 6 ⊢ ((¬ 𝜓 ∧ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
9		andcomm-P3.35 314	. . . . . 6 ⊢ ((¬ 𝜓 ∧ 𝜓) ↔ (𝜓 ∧ ¬ 𝜓))
10	8, 9	subord-P3.43c.RC 351	. . . . 5 ⊢ (((¬ 𝜓 ∧ ¬ 𝜑) ∨ (¬ 𝜓 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜓)))
11		ncontra-P4.1 366	. . . . . 6 ⊢ ¬ (𝜓 ∧ ¬ 𝜓)
12	11	idornthmr-P4.24b 449	. . . . 5 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜓)) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
13	6, 7, 10, 12	tbitrns-P4.17.RC 431	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
14		andcomm-P3.35 314	. . . . 5 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ (¬ 𝜑 ∨ 𝜓)))
15		andoveror-P4.27a 461	. . . . 5 ⊢ ((𝜑 ∧ (¬ 𝜑 ∨ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ 𝜓)))
16		ncontra-P4.1 366	. . . . . 6 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
17	16	idornthml-P4.24a 448	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
18	14, 15, 17	dbitrns-P4.16.RC 429	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
19	13, 18	subord-P3.43c.RC 351	. . 3 ⊢ ((((¬ 𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ∨ ((¬ 𝜑 ∨ 𝜓) ∧ 𝜑)) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
20	1, 4, 5, 19	tbitrns-P4.17.RC 431	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
21		orcomm-P3.37 319	. 2 ⊢ (((¬ 𝜑 ∧ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
22	20, 21	bitrns-P3.33c.RC 303	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132   ∨ wff-or 144

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-false-D2.5 158  df-rcp-AND3 161

This theorem is referenced by: (None)