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Theorem pm5.32d 438
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 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Proof of Theorem pm5.32d
StepHypRef Expression
1 pm5.32d.1 . . . 4 (𝜑 → (𝜓 → (𝜒𝜃)))
2 bi1 116 . . . 4 ((𝜒𝜃) → (𝜒𝜃))
31, 2syl6 33 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
43imdistand 436 . 2 (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
5 bi2 128 . . . 4 ((𝜒𝜃) → (𝜃𝜒))
61, 5syl6 33 . . 3 (𝜑 → (𝜓 → (𝜃𝜒)))
76imdistand 436 . 2 (𝜑 → ((𝜓𝜃) → (𝜓𝜒)))
84, 7impbid 127 1 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm5.32rd  439  pm5.32da  440  pm5.32  441  anbi2d  452  cbvex2  1840  cores  4874  isoini  5508  mpt2eq123  5615  genpassl  6828  genpassu  6829  fzind  8595  btwnz  8599  elfzm11  9236  isprm2  10706  isprm3  10707
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