Theorem syl5com 26

Description: Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)

Hypotheses

Ref	Expression
syl5com.1	⊢ (φ → ψ)
syl5com.2	⊢ (χ → (ψ → θ))

Assertion

Ref	Expression
syl5com	⊢ (φ → (χ → θ))

Proof of Theorem syl5com

Step	Hyp	Ref	Expression
1		syl5com.1	. . 3 ⊢ (φ → ψ)
2	1	a1d 22	. 2 ⊢ (φ → (χ → ψ))
3		syl5com.2	. 2 ⊢ (χ → (ψ → θ))
4	2, 3	sylcom 25	1 ⊢ (φ → (χ → 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:  com12  27  syl5  28  ax16i  2046  ceqsalg  2884  cgsexg  2891  cgsex2g  2892  cgsex4g  2893  spc2egv  2942  spc3egv  2944  disjne  3597  uneqdifeq  3639  ncfinraise  4482  nnpweq  4524  fvimacnv  5404