Theorem bija 344

Description: Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (Contributed by Wolf Lammen, 13-May-2013.)

Hypotheses

Ref	Expression
bija.1	|- (ph -> (ps -> ch))
bija.2	|- (-. ph -> (-. ps -> ch))

Assertion

Ref	Expression
bija	|- ((ph <-> ps) -> ch)

Proof of Theorem bija

Step	Hyp	Ref	Expression
1		bi2 189	. . 3 |- ((ph <-> ps) -> (ps -> ph))
2		bija.1	. . 3 |- (ph -> (ps -> ch))
3	1, 2	syli 33	. 2 |- ((ph <-> ps) -> (ps -> ch))
4		bi1 178	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
5	4	con3d 125	. . 3 |- ((ph <-> ps) -> (-. ps -> -. ph))
6		bija.2	. . 3 |- (-. ph -> (-. ps -> ch))
7	5, 6	syli 33	. 2 |- ((ph <-> ps) -> (-. ps -> ch))
8	3, 7	pm2.61d 150	1 |- ((ph <-> ps) -> ch)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176

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

This theorem is referenced by: (None)