Theorem oroverbi-P4.28b 469

Description: '∨' Distributes Over '↔'.

Assertion

Ref	Expression
oroverbi-P4.28b	⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))

Proof of Theorem oroverbi-P4.28b

Step	Hyp	Ref	Expression
1		oroverbi-P4.28-L3 468	. 2 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))
2		oroverim-P4.28a 467	. . 3 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
3		oroverim-P4.28a 467	. . 3 ⊢ ((𝜑 ∨ (𝜒 → 𝜓)) ↔ ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓)))
4	2, 3	subandd-P3.42c.RC 344	. 2 ⊢ (((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))) ↔ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓))))
5		dfbi-P3.47 358	. . 3 ⊢ (((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)) ↔ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓))))
6	5	bisym-P3.33b.RC 299	. 2 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓))) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))
7	1, 4, 6	dbitrns-P4.16.RC 429	1 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))

Syntax hints:   → 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-rcp-AND3 161  df-rcp-AND4 163

This theorem is referenced by: (None)