Theorem wl-mps 34741

Description: Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
wl-mps.1	⊢ (𝜑 → (𝜓 → 𝜒))
wl-mps.2	⊢ ((𝜑 → 𝜒) → 𝜃)

Assertion

Ref	Expression
wl-mps	⊢ ((𝜑 → 𝜓) → 𝜃)

Proof of Theorem wl-mps

Step	Hyp	Ref	Expression
1		wl-mps.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	a2i 14	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
3		wl-mps.2	. 2 ⊢ ((𝜑 → 𝜒) → 𝜃)
4	2, 3	syl 17	1 ⊢ ((𝜑 → 𝜓) → 𝜃)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  wl-syls1  34742