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Theorem pm5.32d 443
Description: Distribution of implication over biconditional (deduction form). (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 117 . . . 4 ((𝜒𝜃) → (𝜒𝜃))
31, 2syl6 33 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
43imdistand 441 . 2 (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
5 bi2 129 . . . 4 ((𝜒𝜃) → (𝜃𝜒))
61, 5syl6 33 . . 3 (𝜑 → (𝜓 → (𝜃𝜒)))
76imdistand 441 . 2 (𝜑 → ((𝜓𝜃) → (𝜓𝜒)))
84, 7impbid 128 1 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
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:  pm5.32rd  444  pm5.32da  445  pm5.32  446  anbi2d  457  cbvex2  1872  cores  5010  isoini  5685  mpoeq123  5796  genpassl  7296  genpassu  7297  fzind  9120  btwnz  9124  elfzm11  9822  isprm2  11705  isprm3  11706
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