Theorem oplem1 930

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 287	. . . . . 6 |- (-. ps <-> -. th)
3		oplem1.1	. . . . . . 7 |- (ph -> (ps \/ ch))
4	3	ord 366	. . . . . 6 |- (ph -> (-. ps -> ch))
5	2, 4	syl5bir 209	. . . . 5 |- (ph -> (-. th -> ch))
6		oplem1.2	. . . . . 6 |- (ph -> (th \/ ta))
7	6	ord 366	. . . . 5 |- (ph -> (-. th -> ta))
8	5, 7	jcad 519	. . . 4 |- (ph -> (-. th -> (ch /\ ta)))
9		oplem1.4	. . . . 5 |- (ch -> (th <-> ta))
10	9	biimpar 471	. . . 4 |- ((ch /\ ta) -> th)
11	8, 10	syl6 29	. . 3 |- (ph -> (-. th -> th))
12	11	pm2.18d 103	. 2 |- (ph -> th)
13	12, 1	sylibr 203	1 |- (ph -> ps)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

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-or 359  df-an 360

This theorem is referenced by:  preqr1  4125