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Theorem pm5.32 448
 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 445 . 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  449  biadani  601  xordidc  1377  cbvex2  1892  rabbi  2606  rabxfrd  4385  asymref  4919  rexrnmpt  5556  mpo2eqb  5873
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