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Theorem pm5.32 570
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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem pm5.32
StepHypRef Expression
1 notbi 311 . . . 4 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
21imbi2i 328 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)))
3 pm5.74 262 . . 3 ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)))
4 notbi 311 . . 3 (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
52, 3, 43bitri 289 . 2 ((𝜑 → (𝜓𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
6 df-an 386 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
7 df-an 386 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑 → ¬ 𝜒))
86, 7bibi12i 331 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
95, 8bitr4i 270 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385
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 199  df-an 386
This theorem is referenced by:  pm5.32i  571  pm5.32d  573  xordi  1041  rabbi  3302  rabxfrd  5087  asymref  5730  mpt22eqb  7003  cfilucfil4  23447  wl-ax11-lem8  33859  relexp0eq  38776  2sb5nd  39546  2sb5ndVD  39906  2sb5ndALT  39928
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