Theorem biasandorint-P4.34b 492

Description: One direction of '↔' in Terms of '∧' and '∨'. †

Only this direction is deducible with intuitionist logic.

Assertion

Ref	Expression
biasandorint-P4.34b	⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → (𝜑 ↔ 𝜓))

Proof of Theorem biasandorint-P4.34b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . 5 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → (𝜑 ∧ 𝜓))
2	1	ndandel-P3.8 173	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → 𝜓)
3	2	axL1-P3.21 252	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
4	1	ndander-P3.9 174	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → 𝜑)
5	4	axL1-P3.21 252	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
6	3, 5	ndbii-P3.13 178	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))
7		rcp-NDASM2of2 194	. . . . 5 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (¬ 𝜑 ∧ ¬ 𝜓))
8	7	ndander-P3.9 174	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ¬ 𝜑)
9	8	impoe-P4.4a 377	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (𝜑 → 𝜓))
10	7	ndandel-P3.8 173	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ¬ 𝜓)
11	10	impoe-P4.4a 377	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (𝜓 → 𝜑))
12	9, 11	ndbii-P3.13 178	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (𝜑 ↔ 𝜓))
13		rcp-NDASM1of1 192	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
14	6, 12, 13	rcp-NDORE2 235	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

This theorem is referenced by: (None)