Theorem prlem2 1006

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Assertion

Ref	Expression
prlem2	|- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <-> ( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) )

Proof of Theorem prlem2

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( ph /\ ps ) -> ph )
2		simpl 473	. . 3 |- ( ( ch /\ th ) -> ch )
3	1, 2	orim12i 538	. 2 |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) -> ( ph \/ ch ) )
4	3	pm4.71ri 665	1 |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <-> ( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  zfpair  4904