Theorem oplem1 1007

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

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		oplem1.3	. . . . . . 7 |- ( ps <-> th )
2	1	notbii 310	. . . . . 6 |- ( -. ps <-> -. th )
3		oplem1.1	. . . . . . 7 |- ( ph -> ( ps \/ ch ) )
4	3	ord 392	. . . . . 6 |- ( ph -> ( -. ps -> ch ) )
5	2, 4	syl5bir 233	. . . . 5 |- ( ph -> ( -. th -> ch ) )
6		oplem1.2	. . . . . 6 |- ( ph -> ( th \/ ta ) )
7	6	ord 392	. . . . 5 |- ( ph -> ( -. th -> ta ) )
8	5, 7	jcad 555	. . . 4 |- ( ph -> ( -. th -> ( ch /\ ta ) ) )
9		oplem1.4	. . . . 5 |- ( ch -> ( th <-> ta ) )
10	9	biimpar 502	. . . 4 |- ( ( ch /\ ta ) -> th )
11	8, 10	syl6 35	. . 3 |- ( ph -> ( -. th -> th ) )
12	11	pm2.18d 124	. 2 |- ( ph -> th )
13	12, 1	sylibr 224	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  preq1b  4377  preqr1OLD  4380