Theorem pm5.32 578

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

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		notbi 320	. . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
2	1	imbi2i 337	. . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)))
3		pm5.74 271	. . 3 ⊢ ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)))
4		notbi 320	. . 3 ⊢ (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
5	2, 3, 4	3bitri 298	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
6		df-an 397	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
7		df-an 397	. . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜑 → ¬ 𝜒))
8	6, 7	bibi12i 340	. 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
9	5, 8	bitr4i 279	1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 208  df-an 397

This theorem is referenced by:  pm5.32i  579  pm5.32d  582  biadan  824  biadaniALT  826  xordi  1024  rabbi  3422  rabxfrd  5353  asymref  6073  mpo2eqb  7495  cfilucfil4  25313  bj-rcleqf  37385  relexp0eq  44152  2sb5nd  45011  2sb5ndVD  45360  2sb5ndALT  45382  pm5.32dra  49292