Theorem prlem1 973

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 158	. . . 4 |- ( ph -> ( ch -> et ) )
3	2	adantld 278	. . 3 |- ( ph -> ( ( ps /\ ch ) -> et ) )
4		prlem1.2	. . . . 5 |- ( ps -> -. th )
5	4	pm2.21d 619	. . . 4 |- ( ps -> ( th -> et ) )
6	5	adantrd 279	. . 3 |- ( ps -> ( ( th /\ ta ) -> et ) )
7	3, 6	jaao 719	. 2 |- ( ( ph /\ ps ) -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) )
8	7	ex 115	1 |- ( ph -> ( ps -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)