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Theorem bi3ant 217
Description: Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
Hypothesis
Ref Expression
bi3ant.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
bi3ant (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))

Proof of Theorem bi3ant
StepHypRef Expression
1 bi1 115 . . 3 ((𝜃𝜏) → (𝜃𝜏))
21imim1i 58 . 2 (((𝜃𝜏) → 𝜑) → ((𝜃𝜏) → 𝜑))
3 bi2 125 . . 3 ((𝜃𝜏) → (𝜏𝜃))
43imim1i 58 . 2 (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜓))
5 bi3ant.1 . . 3 (𝜑 → (𝜓𝜒))
65imim3i 59 . 2 (((𝜃𝜏) → 𝜑) → (((𝜃𝜏) → 𝜓) → ((𝜃𝜏) → 𝜒)))
72, 4, 6syl2im 38 1 (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  bisym  218
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