Theorem pm5.74 179

Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)

Assertion

Ref	Expression
pm5.74	⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))

Proof of Theorem pm5.74

Step	Hyp	Ref	Expression
1		biimp 118	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
2	1	imim3i 61	. . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
3		biimpr 130	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
4	3	imim3i 61	. . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓)))
5	2, 4	impbid 129	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
6		biimp 118	. . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
7	6	pm2.86d 100	. . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))
8		biimpr 130	. . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓)))
9	8	pm2.86d 100	. . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜒 → 𝜓)))
10	7, 9	impbidd 127	. 2 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
11	5, 10	impbii 126	1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.74i  180  pm5.74ri  181  pm5.74d  182  pm5.74rd  183  bibi2d  232