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	|- (ph -> (ps -> ch))
sylcom.2	|- (ps -> (ch -> th))

Assertion

Ref	Expression
sylcom	|- (ph -> (ps -> th))

Proof of Theorem sylcom

Step	Hyp	Ref	Expression
1		sylcom.1	. 2 |- (ph -> (ps -> ch))
2		sylcom.2	. . 3 |- (ps -> (ch -> th))
3	2	a2i 12	. 2 |- ((ps -> ch) -> (ps -> th))
4	1, 3	syl 15	1 |- (ph -> (ps -> th))

Syntax hints:   -> wi 4

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

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