Theorem imoverbi-P4.30b 479

Description: '→' Left Distributes Over '↔'. †

Assertion

Ref	Expression
imoverbi-P4.30b	⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))

Proof of Theorem imoverbi-P4.30b

Step	Hyp	Ref	Expression
1		imoverbi-P4.30-L2 478	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))))
2		imoverim-P4.30a 477	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
3		imoverim-P4.30a 477	. . 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

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-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  solvesub-P6a  704  lemma-L6.02a  726