Theorem syl3an1 619

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syl3an1.2	|- (ta -> ph)

Assertion

Ref	Expression
syl3an1	|- ((ta /\ ps /\ ch) -> th)

Proof of Theorem syl3an1

Step	Hyp	Ref	Expression
1		syl3an.1	. . . 4 |- ((ph /\ ps /\ ch) -> th)
2	1	3expb 613	. . 3 |- ((ph /\ (ps /\ ch)) -> th)
3		syl3an1.2	. . 3 |- (ta -> ph)
4	2, 3	sylan 343	. 2 |- ((ta /\ (ps /\ ch)) -> th)
5	4	3impb 610	1 |- ((ta /\ ps /\ ch) -> th)

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581

This theorem is referenced by:  syl3an1b 622  syl3an1br 625  nndi 3180  nnmsucr 3182  ecopoprtrn 3247  uzwo3lem1 4614  zbtwnre 4619  projlem26 5218  chlubt 5426  atcvatlem 5770  mdsymlem3 5778  mdsymlem5 5780

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

metamath.org