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Theorem pm5.32 576
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 321 . . . 4 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
21imbi2i 338 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)))
3 pm5.74 272 . . 3 ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)))
4 notbi 321 . . 3 (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
52, 3, 43bitri 299 . 2 ((𝜑 → (𝜓𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
6 df-an 399 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
7 df-an 399 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑 → ¬ 𝜒))
86, 7bibi12i 342 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒)))
95, 8bitr4i 280 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398
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 209  df-an 399
This theorem is referenced by:  pm5.32i  577  pm5.32d  579  biadan  817  biadaniALT  819  xordi  1013  rabbi  3382  cbvrexdva2  3456  rabxfrd  5308  asymref  5969  mpo2eqb  7275  cfilucfil4  23916  bj-rcleqf  34330  wl-ax11-lem8  34816  relexp0eq  40037  2sb5nd  40885  2sb5ndVD  41235  2sb5ndALT  41257
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