Theorem sylcom 25

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)

Hypotheses

Ref	Expression
sylcom.1	⊢ (φ → (ψ → χ))
sylcom.2	⊢ (ψ → (χ → θ))

Assertion

Ref	Expression
sylcom	⊢ (φ → (ψ → θ))

Proof of Theorem sylcom

Step	Hyp	Ref	Expression
1		sylcom.1	. 2 ⊢ (φ → (ψ → χ))
2		sylcom.2	. . 3 ⊢ (ψ → (χ → θ))
3	2	a2i 12	. 2 ⊢ ((ψ → χ) → (ψ → θ))
4	1, 3	syl 15	1 ⊢ (φ → (ψ → θ))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  syl5com  26  syl6  29  syli  33  mpbidi  207