Theorem pm5.32 448

Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)

Assertion

Ref	Expression
pm5.32	⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
2	1	pm5.32d 445	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
3		ibar 299	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
4		ibar 299	. . . 4 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒)))
5	3, 4	bibi12d 234	. . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))))
6	5	biimprcd 159	. 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
7	2, 6	impbii 125	1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.32i  449  biadani  601  xordidc  1377  cbvex2  1892  rabbi  2606  rabxfrd  4385  asymref  4919  rexrnmpt  5556  mpo2eqb  5873