Theorem prlem1 928

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

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	. . . . 5 |- (ph -> (et <-> ch))
2	1	biimprd 214	. . . 4 |- (ph -> (ch -> et))
3	2	adantld 453	. . 3 |- (ph -> ((ps /\ ch) -> et))
4		prlem1.2	. . . . 5 |- (ps -> -. th)
5	4	pm2.21d 98	. . . 4 |- (ps -> (th -> et))
6	5	adantrd 454	. . 3 |- (ps -> ((th /\ ta) -> et))
7	3, 6	jaao 495	. 2 |- ((ph /\ ps) -> (((ps /\ ch) \/ (th /\ ta)) -> et))
8	7	ex 423	1 |- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta)) -> et)))

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: (None)