Theorem prlem1 768

Description: A specialized lemma for set theory (to derive the 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 156	. . . . 5 |- (ch -> (ph -> et))
3	2	adantl 388	. . . 4 |- ((ps /\ ch) -> (ph -> et))
4	3	a1dd 42	. . 3 |- ((ps /\ ch) -> (ph -> (ps -> et)))
5		pm2.24 79	. . . . . 6 |- (th -> (-. th -> et))
6		prlem1.2	. . . . . 6 |- (ps -> -. th)
7	5, 6	syl5 21	. . . . 5 |- (th -> (ps -> et))
8	7	adantr 389	. . . 4 |- ((th /\ ta) -> (ps -> et))
9	8	a1d 12	. . 3 |- ((th /\ ta) -> (ph -> (ps -> et)))
10	4, 9	jaoi 341	. 2 |- (((ps /\ ch) \/ (th /\ ta)) -> (ph -> (ps -> et)))
11	10	com3l 34	1 |- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta)) -> et)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  zfpair 2767

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 147  df-or 224  df-an 225

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