Theorem pm5.32 575

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 319	. . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
2	1	imbi2i 336	. . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)))
3		pm5.74 270	. . 3 ⊢ ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)))
4		notbi 319	. . 3 ⊢ (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
5	2, 3, 4	3bitri 297	. 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
6		df-an 398	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
7		df-an 398	. . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜑 → ¬ 𝜒))
8	6, 7	bibi12i 340	. 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
9	5, 8	bitr4i 278	1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397

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 206  df-an 398

This theorem is referenced by:  pm5.32i  576  pm5.32d  578  biadan  818  biadaniALT  820  xordi  1016  cbvrexdva2OLD  3347  rabbi  3463  rabxfrd  5415  asymref  6115  mpo2eqb  7538  cfilucfil4  24830  bj-rcleqf  35895  wl-ax11-lem8  36443  relexp0eq  42438  2sb5nd  43307  2sb5ndVD  43657  2sb5ndALT  43679  pm5.32dra  47434