Theorem prlem2 577

Description: A specialized lemma for set theory (axiom of pairing).

Assertion

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

Proof of Theorem prlem2

Step	Hyp	Ref	Expression
1		oel 441	. . . . 5 |- (ph <-> ((ph \/ ch) /\ ph))
2	1	anbi1i 368	. . . 4 |- ((ph /\ ps) <-> (((ph \/ ch) /\ ph) /\ ps))
3		anass 336	. . . 4 |- ((((ph \/ ch) /\ ph) /\ ps) <-> ((ph \/ ch) /\ (ph /\ ps)))
4	2, 3	bitr 151	. . 3 |- ((ph /\ ps) <-> ((ph \/ ch) /\ (ph /\ ps)))
5		oel 441	. . . . . 6 |- (ch <-> ((ch \/ ph) /\ ch))
6		orcom 209	. . . . . . 7 |- ((ch \/ ph) <-> (ph \/ ch))
7	6	anbi1i 368	. . . . . 6 |- (((ch \/ ph) /\ ch) <-> ((ph \/ ch) /\ ch))
8	5, 7	bitr 151	. . . . 5 |- (ch <-> ((ph \/ ch) /\ ch))
9	8	anbi1i 368	. . . 4 |- ((ch /\ th) <-> (((ph \/ ch) /\ ch) /\ th))
10		anass 336	. . . 4 |- ((((ph \/ ch) /\ ch) /\ th) <-> ((ph \/ ch) /\ (ch /\ th)))
11	9, 10	bitr 151	. . 3 |- ((ch /\ th) <-> ((ph \/ ch) /\ (ch /\ th)))
12	4, 11	orbi12i 216	. 2 |- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph /\ ps)) \/ ((ph \/ ch) /\ (ch /\ th))))
13		andi 456	. 2 |- (((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))) <-> (((ph \/ ch) /\ (ph /\ ps)) \/ ((ph \/ ch) /\ (ch /\ th))))
14	12, 13	bitr4 154	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))))

Syntax hints:   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  zfpair 1891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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