Theorem acexmidlemph 5881

Description: Lemma for acexmid 5887. (Contributed by Jim Kingdon, 6-Aug-2019.)

Hypotheses

Ref	Expression
acexmidlem.a	|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
acexmidlem.b	|- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
acexmidlem.c	|- C = { A , B }

Assertion

Ref	Expression
acexmidlemph	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   x, C   ph, x

Proof of Theorem acexmidlemph

Step	Hyp	Ref	Expression
1		olc 712	. . . 4 |- ( ph -> ( x = (/) \/ ph ) )
2	1	ralrimivw 2561	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
3		acexmidlem.a	. . . . 5 |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
4	3	eqeq2i 2198	. . . 4 |- ( { (/) , { (/) } } = A <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
5		rabid2 2664	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
6	4, 5	bitri 184	. . 3 |- ( { (/) , { (/) } } = A <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
7	2, 6	sylibr 134	. 2 |- ( ph -> { (/) , { (/) } } = A )
8		olc 712	. . . 4 |- ( ph -> ( x = { (/) } \/ ph ) )
9	8	ralrimivw 2561	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
10		acexmidlem.b	. . . . 5 |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
11	10	eqeq2i 2198	. . . 4 |- ( { (/) , { (/) } } = B <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
12		rabid2 2664	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
13	11, 12	bitri 184	. . 3 |- ( { (/) , { (/) } } = B <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
14	9, 13	sylibr 134	. 2 |- ( ph -> { (/) , { (/) } } = B )
15	7, 14	eqtr3d 2222	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1363   A.wral 2465   {crab 2469   (/)c0 3434   {csn 3604   {cpr 3605

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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-rab 2474

This theorem is referenced by:  acexmidlemab  5882