Theorem acexmidlemph 5659

Description: Lemma for acexmid 5665. (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 668	. . . 4 |- ( ph -> ( x = (/) \/ ph ) )
2	1	ralrimivw 2448	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
3		acexmidlem.a	. . . . 5 |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
4	3	eqeq2i 2099	. . . 4 |- ( { (/) , { (/) } } = A <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
5		rabid2 2544	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
6	4, 5	bitri 183	. . 3 |- ( { (/) , { (/) } } = A <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
7	2, 6	sylibr 133	. 2 |- ( ph -> { (/) , { (/) } } = A )
8		olc 668	. . . 4 |- ( ph -> ( x = { (/) } \/ ph ) )
9	8	ralrimivw 2448	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
10		acexmidlem.b	. . . . 5 |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
11	10	eqeq2i 2099	. . . 4 |- ( { (/) , { (/) } } = B <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
12		rabid2 2544	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
13	11, 12	bitri 183	. . 3 |- ( { (/) , { (/) } } = B <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
14	9, 13	sylibr 133	. 2 |- ( ph -> { (/) , { (/) } } = B )
15	7, 14	eqtr3d 2123	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   \/ wo 665   = wceq 1290   A.wral 2360   {crab 2364   (/)c0 3287   {csn 3450   {cpr 3451

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-rab 2369

This theorem is referenced by:  acexmidlemab  5660