Theorem eqerlem 6532

Description: Lemma for eqer 6533. (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 4239	. 2 |- ( z R w <-> [. z / x ]. [. w / y ]. A = B )
3		vex 2729	. . 3 |- z e. _V
4		nfcsb1v 3078	. . . . 5 |- F/_ x [_ z / x ]_ A
5		nfcsb1v 3078	. . . . 5 |- F/_ x [_ w / x ]_ A
6	4, 5	nfeq 2316	. . . 4 |- F/ x [_ z / x ]_ A = [_ w / x ]_ A
7		vex 2729	. . . . . 6 |- w e. _V
8		nfv 1516	. . . . . . 7 |- F/ y A = [_ w / x ]_ A
9		vex 2729	. . . . . . . . . 10 |- y e. _V
10		nfcv 2308	. . . . . . . . . 10 |- F/_ x B
11		eqer.1	. . . . . . . . . 10 |- ( x = y -> A = B )
12	9, 10, 11	csbief 3089	. . . . . . . . 9 |- [_ y / x ]_ A = B
13		csbeq1 3048	. . . . . . . . 9 |- ( y = w -> [_ y / x ]_ A = [_ w / x ]_ A )
14	12, 13	eqtr3id 2213	. . . . . . . 8 |- ( y = w -> B = [_ w / x ]_ A )
15	14	eqeq2d 2177	. . . . . . 7 |- ( y = w -> ( A = B <-> A = [_ w / x ]_ A ) )
16	8, 15	sbciegf 2982	. . . . . 6 |- ( w e. _V -> ( [. w / y ]. A = B <-> A = [_ w / x ]_ A ) )
17	7, 16	ax-mp 5	. . . . 5 |- ( [. w / y ]. A = B <-> A = [_ w / x ]_ A )
18		csbeq1a 3054	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
19	18	eqeq1d 2174	. . . . 5 |- ( x = z -> ( A = [_ w / x ]_ A <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
20	17, 19	syl5bb 191	. . . 4 |- ( x = z -> ( [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
21	6, 20	sbciegf 2982	. . 3 |- ( z e. _V -> ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
22	3, 21	ax-mp 5	. 2 |- ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A )
23	2, 22	bitri 183	1 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   [.wsbc 2951   [_csb 3045   class class class wbr 3982   {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044

This theorem is referenced by:  eqer  6533