Theorem syl3an1b 1133

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syl3an1b.2	|- (ta <-> ph)

Assertion

Ref	Expression
syl3an1b	|- ((ta /\ ps /\ ch) -> th)

Proof of Theorem syl3an1b

Step	Hyp	Ref	Expression
1		syl3an.1	. 2 |- ((ph /\ ps /\ ch) -> th)
2		syl3an1b.2	. . 3 |- (ta <-> ph)
3	2	biimpi 168	. 2 |- (ta -> ph)
4	1, 3	syl3an1 1130	1 |- ((ta /\ ps /\ ch) -> th)

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858

This theorem is referenced by:  xrlttr 6728  climcmplem 8397  seq0cl 13620  fgsb 14921  fgsb2 14925  wofi 15770  divrngcl 16110

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860

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