Theorem prlem2 771

Description: A specialized lemma for set theory (to derive the 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		orabs 581	. . . . 5 |- (ph <-> ((ph \/ ch) /\ ph))
2	1	anbi1i 481	. . . 4 |- ((ph /\ ps) <-> (((ph \/ ch) /\ ph) /\ ps))
3		anass 439	. . . 4 |- ((((ph \/ ch) /\ ph) /\ ps) <-> ((ph \/ ch) /\ (ph /\ ps)))
4	2, 3	bitr 173	. . 3 |- ((ph /\ ps) <-> ((ph \/ ch) /\ (ph /\ ps)))
5		orabs 581	. . . . . 6 |- (ch <-> ((ch \/ ph) /\ ch))
6		orcom 246	. . . . . . 7 |- ((ch \/ ph) <-> (ph \/ ch))
7	6	anbi1i 481	. . . . . 6 |- (((ch \/ ph) /\ ch) <-> ((ph \/ ch) /\ ch))
8	5, 7	bitr 173	. . . . 5 |- (ch <-> ((ph \/ ch) /\ ch))
9	8	anbi1i 481	. . . 4 |- ((ch /\ th) <-> (((ph \/ ch) /\ ch) /\ th))
10		anass 439	. . . 4 |- ((((ph \/ ch) /\ ch) /\ th) <-> ((ph \/ ch) /\ (ch /\ th)))
11	9, 10	bitr 173	. . 3 |- ((ch /\ th) <-> ((ph \/ ch) /\ (ch /\ th)))
12	4, 11	orbi12i 257	. 2 |- (((ph /\ ps) \/ (ch /\ th)) <-> (((ph \/ ch) /\ (ph /\ ps)) \/ ((ph \/ ch) /\ (ch /\ th))))
13		andi 604	. 2 |- (((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))) <-> (((ph \/ ch) /\ (ph /\ ps)) \/ ((ph \/ ch) /\ (ch /\ th))))
14	12, 13	bitr4 176	1 |- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  zfpair 2777

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 147  df-or 224  df-an 225

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