Theorem syl7bi 164

Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
syl7bi.1	|- ( ph <-> ps )
syl7bi.2	|- ( ch -> ( th -> ( ps -> ta ) ) )

Assertion

Ref	Expression
syl7bi	|- ( ch -> ( th -> ( ph -> ta ) ) )

Proof of Theorem syl7bi

Step	Hyp	Ref	Expression
1		syl7bi.1	. . 3 |- ( ph <-> ps )
2	1	biimpi 119	. 2 |- ( ph -> ps )
3		syl7bi.2	. 2 |- ( ch -> ( th -> ( ps -> ta ) ) )
4	2, 3	syl7 69	1 |- ( ch -> ( th -> ( ph -> ta ) ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  necon1addc  2416  necon1ddc  2418  rspct  2827  2reuswapdc  2934  nn0lt2  9293  fzofzim  10144  ndvdssub  11889  bj-findis  14014