Theorem mpan10 457

Description: Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)

Assertion

Ref	Expression
mpan10	⊢ ((((𝜑 → 𝜓) ∧ 𝜒) ∧ 𝜑) → (𝜓 ∧ 𝜒))

Proof of Theorem mpan10

Step	Hyp	Ref	Expression
1		ancom 262	. . . 4 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒))
2	1	anbi2i 444	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ∧ 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∧ 𝜒)))
3		anass 393	. . 3 ⊢ ((((𝜑 → 𝜓) ∧ 𝜒) ∧ 𝜑) ↔ ((𝜑 → 𝜓) ∧ (𝜒 ∧ 𝜑)))
4		anass 393	. . 3 ⊢ ((((𝜑 → 𝜓) ∧ 𝜑) ∧ 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∧ 𝜒)))
5	2, 3, 4	3bitr4i 210	. 2 ⊢ ((((𝜑 → 𝜓) ∧ 𝜒) ∧ 𝜑) ↔ (((𝜑 → 𝜓) ∧ 𝜑) ∧ 𝜒))
6		id 19	. . . 4 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
7	6	imp 122	. . 3 ⊢ (((𝜑 → 𝜓) ∧ 𝜑) → 𝜓)
8	7	anim1i 333	. 2 ⊢ ((((𝜑 → 𝜓) ∧ 𝜑) ∧ 𝜒) → (𝜓 ∧ 𝜒))
9	5, 8	sylbi 119	1 ⊢ ((((𝜑 → 𝜓) ∧ 𝜒) ∧ 𝜑) → (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)