Theorem eqerlem 7776

Description: Lemma for eqer 7777. (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 4986	. 2 |- ( z R w <-> [. z / x ]. [. w / y ]. A = B )
3		vex 3203	. . 3 |- z e. _V
4		nfcsb1v 3549	. . . . 5 |- F/_ x [_ z / x ]_ A
5		nfcsb1v 3549	. . . . 5 |- F/_ x [_ w / x ]_ A
6	4, 5	nfeq 2776	. . . 4 |- F/ x [_ z / x ]_ A = [_ w / x ]_ A
7		vex 3203	. . . . . 6 |- w e. _V
8		nfv 1843	. . . . . . 7 |- F/ y A = [_ w / x ]_ A
9		vex 3203	. . . . . . . . . 10 |- y e. _V
10		eqer.1	. . . . . . . . . 10 |- ( x = y -> A = B )
11	9, 10	csbie 3559	. . . . . . . . 9 |- [_ y / x ]_ A = B
12		csbeq1 3536	. . . . . . . . 9 |- ( y = w -> [_ y / x ]_ A = [_ w / x ]_ A )
13	11, 12	syl5eqr 2670	. . . . . . . 8 |- ( y = w -> B = [_ w / x ]_ A )
14	13	eqeq2d 2632	. . . . . . 7 |- ( y = w -> ( A = B <-> A = [_ w / x ]_ A ) )
15	8, 14	sbciegf 3467	. . . . . 6 |- ( w e. _V -> ( [. w / y ]. A = B <-> A = [_ w / x ]_ A ) )
16	7, 15	ax-mp 5	. . . . 5 |- ( [. w / y ]. A = B <-> A = [_ w / x ]_ A )
17		csbeq1a 3542	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
18	17	eqeq1d 2624	. . . . 5 |- ( x = z -> ( A = [_ w / x ]_ A <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
19	16, 18	syl5bb 272	. . . 4 |- ( x = z -> ( [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
20	6, 19	sbciegf 3467	. . 3 |- ( z e. _V -> ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
21	3, 20	ax-mp 5	. 2 |- ( [. z / x ]. [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A )
22	2, 21	bitri 264	1 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   class class class wbr 4653   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  eqer  7777  eqerOLD  7778