Theorem sylcom 51

Description: Syllogism inference with commutation of antecedents.

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	. . . 4 ⊢ (φ → (ψ → χ))
2	1	com12 13	. . 3 ⊢ (ψ → (φ → χ))
3		sylcom.2	. . 3 ⊢ (ψ → (χ → θ))
4	2, 3	syld 27	. 2 ⊢ (ψ → (φ → θ))
5	4	com12 13	1 ⊢ (φ → (ψ → θ))

Syntax hints:   → wi 2

This theorem is referenced by:  syli 52  abianfp 3000  unblem3 3433  isfinite2 3437  nsmallpq 3877  uzwo 4605  nnwoOLD 4608  chcmh 5148  h1datom 5483  pjjs 5585

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-mp 6

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