Theorem oplem1 578

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 202	. . . 4 |- (ph -> (-. ps -> ch))
3		oplem1.2	. . . . . 6 |- (ph -> (th \/ ta ))
4	3	ord 202	. . . . 5 |- (ph -> (-. th -> ta ))
5		oplem1.3	. . . . . 6 |- (ps <-> th)
6	5	negbii 162	. . . . 5 |- (-. ps <-> -. th)
7	4, 6	syl5ib 181	. . . 4 |- (ph -> (-. ps -> ta ))
8	2, 7	jcad 455	. . 3 |- (ph -> (-. ps -> (ch /\ ta )))
9		oplem1.4	. . . . 5 |- (ch -> (th <-> ta ))
10	9, 5	syl5bb 410	. . . 4 |- (ch -> (ps <-> ta ))
11	10	biimpar 325	. . 3 |- ((ch /\ ta ) -> ps)
12	8, 11	syl6 23	. 2 |- (ph -> (-. ps -> ps))
13		pm2.18 75	. 2 |- ((-. ps -> ps) -> ps)
14	12, 13	syl 12	1 |- (ph -> ps)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  preqr1 1872

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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