Theorem eqerlem 6160

Description: Lemma for eqer 6161. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypotheses

Ref	Expression
eqer.1	|- ( x = y -> A = B )
eqer.2	|- R = { <. x , y >. | A = B }

Assertion

Ref	Expression
eqerlem	|- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )

Distinct variable groups:   x, w, y   x, z, y   y, A   x, B

Allowed substitution hints:   A( x, z, w)   B( y, z, w)   R( x, y, z, w)

Proof of Theorem eqerlem

Step	Hyp	Ref	Expression
1		eqer.2	. . 3 |- R = { <. x , y >. | A = B }
2	1	brabsb 4016	. 2 |- ( z R w <-> [. z / x ]. [. w / y ]. A = B )
3		vex 2604	. . 3 |- z e. _V
4		nfcsb1v 2938	. . . . 5 |- F/_ x [_ z / x ]_ A
5		nfcsb1v 2938	. . . . 5 |- F/_ x [_ w / x ]_ A
6	4, 5	nfeq 2226	. . . 4 |- F/ x [_ z / x ]_ A = [_ w / x ]_ A
7		vex 2604	. . . . . 6 |- w e. _V
8		nfv 1461	. . . . . . 7 |- F/ y A = [_ w / x ]_ A
9		vex 2604	. . . . . . . . . 10 |- y e. _V
10		nfcv 2219	. . . . . . . . . 10 |- F/_ x B
11		eqer.1	. . . . . . . . . 10 |- ( x = y -> A = B )
12	9, 10, 11	csbief 2947	. . . . . . . . 9 |- [_ y / x ]_ A = B
13		csbeq1 2911	. . . . . . . . 9 |- ( y = w -> [_ y / x ]_ A = [_ w / x ]_ A )
14	12, 13	syl5eqr 2127	. . . . . . . 8 |- ( y = w -> B = [_ w / x ]_ A )
15	14	eqeq2d 2092	. . . . . . 7 |- ( y = w -> ( A = B <-> A = [_ w / x ]_ A ) )
16	8, 15	sbciegf 2845	. . . . . 6 |- ( w e. _V -> ( [. w / y ]. A = B <-> A = [_ w / x ]_ A ) )
17	7, 16	ax-mp 7	. . . . 5 |- ( [. w / y ]. A = B <-> A = [_ w / x ]_ A )
18		csbeq1a 2916	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
19	18	eqeq1d 2089	. . . . 5 |- ( x = z -> ( A = [_ w / x ]_ A <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
20	17, 19	syl5bb 190	. . . 4 |- ( x = z -> ( [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
21	6, 20	sbciegf 2845	. . 3 |- ( z e. _V -> ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
22	3, 21	ax-mp 7	. 2 |- ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A )
23	2, 22	bitri 182	1 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   [.wsbc 2815   [_csb 2908   class class class wbr 3785   {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840

This theorem is referenced by:  eqer  6161