Theorem acexmidlemph 5505

Description: Lemma for acexmid 5511. (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 632	. . . 4 |- ( ph -> ( x = (/) \/ ph ) )
2	1	ralrimivw 2393	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
3		acexmidlem.a	. . . . 5 |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
4	3	eqeq2i 2050	. . . 4 |- ( { (/) , { (/) } } = A <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
5		rabid2 2486	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
6	4, 5	bitri 173	. . 3 |- ( { (/) , { (/) } } = A <-> A. x e. { (/) , { (/) } } ( x = (/) \/ ph ) )
7	2, 6	sylibr 137	. 2 |- ( ph -> { (/) , { (/) } } = A )
8		olc 632	. . . 4 |- ( ph -> ( x = { (/) } \/ ph ) )
9	8	ralrimivw 2393	. . 3 |- ( ph -> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
10		acexmidlem.b	. . . . 5 |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }
11	10	eqeq2i 2050	. . . 4 |- ( { (/) , { (/) } } = B <-> { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } )
12		rabid2 2486	. . . 4 |- ( { (/) , { (/) } } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) } <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
13	11, 12	bitri 173	. . 3 |- ( { (/) , { (/) } } = B <-> A. x e. { (/) , { (/) } } ( x = { (/) } \/ ph ) )
14	9, 13	sylibr 137	. 2 |- ( ph -> { (/) , { (/) } } = B )
15	7, 14	eqtr3d 2074	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   A.wral 2306   {crab 2310   (/)c0 3224   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rab 2315

This theorem is referenced by:  acexmidlemab  5506