Theorem oplem1 977

Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Hypotheses

Ref	Expression
oplem1.1	|- ( ph -> ( ps \/ ch ) )
oplem1.2	|- ( ph -> ( th \/ ta ) )
oplem1.3	|- ( ps <-> th )
oplem1.4	|- ( ch -> ( th <-> ta ) )

Assertion

Ref	Expression
oplem1	|- ( ph -> ps )

Proof of Theorem oplem1

Step	Hyp	Ref	Expression
1		oplem1.1	. 2 |- ( ph -> ( ps \/ ch ) )
2		idd 21	. . 3 |- ( ph -> ( ps -> ps ) )
3		oplem1.2	. . . . 5 |- ( ph -> ( th \/ ta ) )
4		ax-1 6	. . . . . 6 |- ( th -> ( ch -> th ) )
5		oplem1.4	. . . . . . 7 |- ( ch -> ( th <-> ta ) )
6	5	biimprcd 160	. . . . . 6 |- ( ta -> ( ch -> th ) )
7	4, 6	jaoi 717	. . . . 5 |- ( ( th \/ ta ) -> ( ch -> th ) )
8	3, 7	syl 14	. . . 4 |- ( ph -> ( ch -> th ) )
9		oplem1.3	. . . 4 |- ( ps <-> th )
10	8, 9	imbitrrdi 162	. . 3 |- ( ph -> ( ch -> ps ) )
11	2, 10	jaod 718	. 2 |- ( ph -> ( ( ps \/ ch ) -> ps ) )
12	1, 11	mpd 13	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  preqr1g  3797  preqr1  3799