Theorem prlem1 576

Description: A specialized lemma for set theory (axiom of pairing).

Hypotheses

Ref	Expression
prlem1.1	|- (ph -> (et <-> ch))
prlem1.2	|- (ps -> -. th)

Assertion

Ref	Expression
prlem1	|- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta )) -> et)))

Proof of Theorem prlem1

Step	Hyp	Ref	Expression
1		prlem1.1	. . . . . 6 |- (ph -> (et <-> ch))
2	1	biimprcd 138	. . . . 5 |- (ch -> (ph -> et))
3	2	adantl 305	. . . 4 |- ((ps /\ ch) -> (ph -> et))
4	3	a1dd 42	. . 3 |- ((ps /\ ch) -> (ph -> (ps -> et)))
5		pm2.24 72	. . . . . 6 |- (th -> (-. th -> et))
6		prlem1.2	. . . . . 6 |- (ps -> -. th)
7	5, 6	syl5 22	. . . . 5 |- (th -> (ps -> et))
8	7	adantr 306	. . . 4 |- ((th /\ ta ) -> (ps -> et))
9	8	a1d 14	. . 3 |- ((th /\ ta ) -> (ph -> (ps -> et)))
10	4, 9	jaoi 275	. 2 |- (((ps /\ ch) \/ (th /\ ta )) -> (ph -> (ps -> et)))
11	10	com3l 34	1 |- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta )) -> et)))

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

This theorem is referenced by:  zfpair 1891

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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