Theorem imoverbi-P4.30-L2 478

Description: Lemma for imoverbi-P4.30b 479. †

Assertion

Ref	Expression
imoverbi-P4.30-L2	⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → (𝜓 → 𝜒)) ∧ (𝜑 → (𝜒 → 𝜓))))

Proof of Theorem imoverbi-P4.30-L2

Step	Hyp	Ref	Expression
1		dfbi-P3.47 358	. . 3 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
2	1	subimr-P3.40b.RC 328	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))))
3		imoverand-P4.29a 472	. 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

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

This theorem is referenced by:  imoverbi-P4.30b  479