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Theorem pm5.32d 423
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
Hypothesis
Ref Expression
pm5.32d.1 (φ → (ψ → (χθ)))
Assertion
Ref Expression
pm5.32d (φ → ((ψ χ) ↔ (ψ θ)))

Proof of Theorem pm5.32d
StepHypRef Expression
1 pm5.32d.1 . . . 4 (φ → (ψ → (χθ)))
2 bi1 111 . . . 4 ((χθ) → (χθ))
31, 2syl6 29 . . 3 (φ → (ψ → (χθ)))
43imdistand 421 . 2 (φ → ((ψ χ) → (ψ θ)))
5 bi2 121 . . . 4 ((χθ) → (θχ))
61, 5syl6 29 . . 3 (φ → (ψ → (θχ)))
76imdistand 421 . 2 (φ → ((ψ θ) → (ψ χ)))
84, 7impbid 120 1 (φ → ((ψ χ) ↔ (ψ θ)))
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:  pm5.32rd  424  pm5.32da  425  pm5.32  426  anbi2d  437  cbvex2  1794  cores  4767  isoini  5400  mpt2eq123  5506  genpassl  6507  genpassu  6508  fzind  8129  btwnz  8133  elfzm11  8723
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