Theorem eqerlem 6565

Description: Lemma for eqer 6566. (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 4261	. 2 |- ( z R w <-> [. z / x ]. [. w / y ]. A = B )
3		vex 2740	. . 3 |- z e. _V
4		nfcsb1v 3090	. . . . 5 |- F/_ x [_ z / x ]_ A
5		nfcsb1v 3090	. . . . 5 |- F/_ x [_ w / x ]_ A
6	4, 5	nfeq 2327	. . . 4 |- F/ x [_ z / x ]_ A = [_ w / x ]_ A
7		vex 2740	. . . . . 6 |- w e. _V
8		nfv 1528	. . . . . . 7 |- F/ y A = [_ w / x ]_ A
9		vex 2740	. . . . . . . . . 10 |- y e. _V
10		nfcv 2319	. . . . . . . . . 10 |- F/_ x B
11		eqer.1	. . . . . . . . . 10 |- ( x = y -> A = B )
12	9, 10, 11	csbief 3101	. . . . . . . . 9 |- [_ y / x ]_ A = B
13		csbeq1 3060	. . . . . . . . 9 |- ( y = w -> [_ y / x ]_ A = [_ w / x ]_ A )
14	12, 13	eqtr3id 2224	. . . . . . . 8 |- ( y = w -> B = [_ w / x ]_ A )
15	14	eqeq2d 2189	. . . . . . 7 |- ( y = w -> ( A = B <-> A = [_ w / x ]_ A ) )
16	8, 15	sbciegf 2994	. . . . . 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 3066	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
19	18	eqeq1d 2186	. . . . 5 |- ( x = z -> ( A = [_ w / x ]_ A <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
20	17, 19	bitrid 192	. . . 4 |- ( x = z -> ( [. w / y ]. A = B <-> [_ z / x ]_ A = [_ w / x ]_ A ) )
21	6, 20	sbciegf 2994	. . 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 184	1 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   [.wsbc 2962   [_csb 3057   class class class wbr 4003   {copab 4063

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065

This theorem is referenced by:  eqer  6566