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Theorem mpan10 443
Description: Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
Assertion
Ref Expression
mpan10 ((((φψ) χ) φ) → (ψ χ))

Proof of Theorem mpan10
StepHypRef Expression
1 ancom 253 . . . 4 ((χ φ) ↔ (φ χ))
21anbi2i 430 . . 3 (((φψ) (χ φ)) ↔ ((φψ) (φ χ)))
3 anass 381 . . 3 ((((φψ) χ) φ) ↔ ((φψ) (χ φ)))
4 anass 381 . . 3 ((((φψ) φ) χ) ↔ ((φψ) (φ χ)))
52, 3, 43bitr4i 201 . 2 ((((φψ) χ) φ) ↔ (((φψ) φ) χ))
6 id 19 . . . 4 ((φψ) → (φψ))
76imp 115 . . 3 (((φψ) φ) → ψ)
87anim1i 323 . 2 ((((φψ) φ) χ) → (ψ χ))
95, 8sylbi 114 1 ((((φψ) χ) φ) → (ψ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97
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: (None)
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