Theorem acexmidlemph 6043

Description: Lemma for acexmid 6049. (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 719	. . . 4 |- ( ph -> ( x = (/) \/ ph ) )
2	1	ralrimivw 2616	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
3		acexmidlem.a	. . . . 5 |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
4	3	eqeq2i 2243	. . . 4 |- ( { (/) , { (/) } } = A <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
5		rabid2 2721	. . . 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 719	. . . 4 |- ( ph -> ( x = { (/) } \/ ph ) )
9	8	ralrimivw 2616	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
10		acexmidlem.b	. . . . 5 |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
11	10	eqeq2i 2243	. . . 4 |- ( { (/) , { (/) } } = B <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
12		rabid2 2721	. . . 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 2267	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   A.wral 2520   {crab 2524   (/)c0 3508   {csn 3689   {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-rab 2529

This theorem is referenced by:  acexmidlemab  6044