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Theorem pm5.32 426
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)) ↔ ((φ ψ) ↔ (φ χ)))

Proof of Theorem pm5.32
StepHypRef Expression
1 id 19 . . 3 ((φ → (ψχ)) → (φ → (ψχ)))
21pm5.32d 423 . 2 ((φ → (ψχ)) → ((φ ψ) ↔ (φ χ)))
3 ibar 285 . . . 4 (φ → (ψ ↔ (φ ψ)))
4 ibar 285 . . . 4 (φ → (χ ↔ (φ χ)))
53, 4bibi12d 224 . . 3 (φ → ((ψχ) ↔ ((φ ψ) ↔ (φ χ))))
65biimprcd 149 . 2 (((φ ψ) ↔ (φ χ)) → (φ → (ψχ)))
72, 6impbii 117 1 ((φ → (ψχ)) ↔ ((φ ψ) ↔ (φ χ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm5.32i  427  xordidc  1287  cbvex2  1794  rabbi  2481  rabxfrd  4167  asymref  4653  rexrnmpt  5253  mpt22eqb  5552
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