Theorem prlem1 848

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (The proof was shortened by Andrew Salmon, 13-May-2011.)

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	biimprd 171	. . . . 5 |- (ph -> (ch -> et))
3	2	adantld 426	. . . 4 |- (ph -> ((ps /\ ch) -> et))
4	3	adantr 425	. . 3 |- ((ph /\ ps) -> ((ps /\ ch) -> et))
5		prlem1.2	. . . . . 6 |- (ps -> -. th)
6	5	pm2.21d 94	. . . . 5 |- (ps -> (th -> et))
7	6	adantrd 427	. . . 4 |- (ps -> ((th /\ ta) -> et))
8	7	adantl 424	. . 3 |- ((ph /\ ps) -> ((th /\ ta) -> et))
9	4, 8	jaod 469	. 2 |- ((ph /\ ps) -> (((ps /\ ch) \/ (th /\ ta)) -> et))
10	9	ex 402	1 |- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta)) -> et)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240

This theorem is referenced by:  zfpair 3522

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 164  df-or 241  df-an 242

Copyright terms: Public domain