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Theorem pm5.32 449
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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem pm5.32
StepHypRef Expression
1 id 19 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓𝜒)))
21pm5.32d 446 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
3 ibar 299 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
4 ibar 299 . . . 4 (𝜑 → (𝜒 ↔ (𝜑𝜒)))
53, 4bibi12d 234 . . 3 (𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
65biimprcd 159 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
72, 6impbii 125 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff set class
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  450  biadani  602  xordidc  1388  cbvex2  1909  rabbi  2641  rabxfrd  4442  asymref  4984  rexrnmpt  5623  mpo2eqb  5943
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