Theorem syl3anbrc 1136

Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)

Hypotheses

Ref	Expression
syl3anbrc.1	|- (ph -> ps)
syl3anbrc.2	|- (ph -> ch)
syl3anbrc.3	|- (ph -> th)
syl3anbrc.4	|- (ta <-> (ps /\ ch /\ th))

Assertion

Ref	Expression
syl3anbrc	|- (ph -> ta)

Proof of Theorem syl3anbrc

Step	Hyp	Ref	Expression
1		syl3anbrc.1	. . 3 |- (ph -> ps)
2		syl3anbrc.2	. . 3 |- (ph -> ch)
3		syl3anbrc.3	. . 3 |- (ph -> th)
4	1, 2, 3	3jca 1132	. 2 |- (ph -> (ps /\ ch /\ th))
5		syl3anbrc.4	. 2 |- (ta <-> (ps /\ ch /\ th))
6	4, 5	sylibr 203	1 |- (ph -> ta)

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by:  sfindbl  4530  vfinspsslem1  4550  pod  5936  fundmen  6043  enmap1  6074  enprmap  6082