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Theorem mpan10 451
Description: Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
Assertion
Ref Expression
mpan10 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜑) → (𝜓𝜒))

Proof of Theorem mpan10
StepHypRef Expression
1 ancom 257 . . . 4 ((𝜒𝜑) ↔ (𝜑𝜒))
21anbi2i 438 . . 3 (((𝜑𝜓) ∧ (𝜒𝜑)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
3 anass 387 . . 3 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜑) ↔ ((𝜑𝜓) ∧ (𝜒𝜑)))
4 anass 387 . . 3 ((((𝜑𝜓) ∧ 𝜑) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
52, 3, 43bitr4i 205 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜑) ↔ (((𝜑𝜓) ∧ 𝜑) ∧ 𝜒))
6 id 19 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
76imp 119 . . 3 (((𝜑𝜓) ∧ 𝜑) → 𝜓)
87anim1i 327 . 2 ((((𝜑𝜓) ∧ 𝜑) ∧ 𝜒) → (𝜓𝜒))
95, 8sylbi 118 1 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜑) → (𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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