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Theorem bi3ant 213
 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 111 . . 3 ((θτ) → (θτ))
21imim1i 54 . 2 (((θτ) → φ) → ((θτ) → φ))
3 bi2 121 . . 3 ((θτ) → (τθ))
43imim1i 54 . 2 (((τθ) → ψ) → ((θτ) → ψ))
5 bi3ant.1 . . 3 (φ → (ψχ))
65imim3i 55 . 2 (((θτ) → φ) → (((θτ) → ψ) → ((θτ) → χ)))
72, 4, 6syl2im 34 1 (((θτ) → φ) → (((τθ) → ψ) → ((θτ) → χ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  bisym  214
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