Theorem prlem2 850

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

Assertion

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

Proof of Theorem prlem2

Step	Hyp	Ref	Expression
1		orc 291	. . . . 5 |- (ph -> (ph \/ ch))
2	1	adantr 425	. . . 4 |- ((ph /\ ps) -> (ph \/ ch))
3		olc 290	. . . . 5 |- (ch -> (ph \/ ch))
4	3	adantr 425	. . . 4 |- ((ch /\ th) -> (ph \/ ch))
5	2, 4	jaoi 368	. . 3 |- (((ph /\ ps) \/ (ch /\ th)) -> (ph \/ ch))
6		id 73	. . 3 |- (((ph /\ ps) \/ (ch /\ th)) -> ((ph /\ ps) \/ (ch /\ th)))
7	5, 6	jca 310	. 2 |- (((ph /\ ps) \/ (ch /\ th)) -> ((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))))
8		simpr 350	. 2 |- (((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))) -> ((ph /\ ps) \/ (ch /\ th)))
9	7, 8	impbii 174	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))))

Syntax hints:   <-> 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

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