Theorem oroverbi-P4.28-L3 468

Description: Lemma for oroverbi-P4.28b 469 and oroverbiint-P4.28d 471. †

Assertion

Ref	Expression
oroverbi-P4.28-L3	⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))

Proof of Theorem oroverbi-P4.28-L3

Step	Hyp	Ref	Expression
1		dfbi-P3.47 358	. . 3 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
2	1	suborr-P3.43b.RC 349	. 2 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ (𝜑 ∨ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))))
3		oroverand-P4.27b 464	. 2 ⊢ ((𝜑 ∨ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))) ↔ ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))
4	2, 3	bitrns-P3.33c.RC 303	1 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))

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:  oroverbi-P4.28b  469  oroverbiint-P4.28d  471