Theorem oplem1 784

Description: A specialized lemma for set theory (ordered pair theorem).

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	. . . . 5 |- (ph -> (ps \/ ch))
2	1	ord 239	. . . 4 |- (ph -> (-. ps -> ch))
3		oplem1.2	. . . . . 6 |- (ph -> (th \/ ta))
4	3	ord 239	. . . . 5 |- (ph -> (-. th -> ta))
5		oplem1.3	. . . . . 6 |- (ps <-> th)
6	5	notbii 194	. . . . 5 |- (-. ps <-> -. th)
7	4, 6	syl5ib 213	. . . 4 |- (ph -> (-. ps -> ta))
8	2, 7	jcad 611	. . 3 |- (ph -> (-. ps -> (ch /\ ta)))
9		oplem1.4	. . . . 5 |- (ch -> (th <-> ta))
10	9, 5	syl5bb 543	. . . 4 |- (ch -> (ps <-> ta))
11	10	biimpar 426	. . 3 |- ((ch /\ ta) -> ps)
12	8, 11	syl6 22	. 2 |- (ph -> (-. ps -> ps))
13		pm2.18 84	. 2 |- ((-. ps -> ps) -> ps)
14	12, 13	syl 10	1 |- (ph -> ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   \/ wo 229   /\ wa 230

This theorem is referenced by:  preqr1 2535

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232

Copyright terms: Public domain