Theorem eqerlem 6544

Description: Lemma for eqer 6545. (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 4246	. 2 |- ( z R w <-> [. z / x ]. [. w / y ]. A = B )
3		vex 2733	. . 3 |- z e. _V
4		nfcsb1v 3082	. . . . 5 |- F/_ x [_ z / x ]_ A
5		nfcsb1v 3082	. . . . 5 |- F/_ x [_ w / x ]_ A
6	4, 5	nfeq 2320	. . . 4 |- F/ x [_ z / x ]_ A = [_ w / x ]_ A
7		vex 2733	. . . . . 6 |- w e. _V
8		nfv 1521	. . . . . . 7 |- F/ y A = [_ w / x ]_ A
9		vex 2733	. . . . . . . . . 10 |- y e. _V
10		nfcv 2312	. . . . . . . . . 10 |- F/_ x B
11		eqer.1	. . . . . . . . . 10 |- ( x = y -> A = B )
12	9, 10, 11	csbief 3093	. . . . . . . . 9 |- [_ y / x ]_ A = B
13		csbeq1 3052	. . . . . . . . 9 |- ( y = w -> [_ y / x ]_ A = [_ w / x ]_ A )
14	12, 13	eqtr3id 2217	. . . . . . . 8 |- ( y = w -> B = [_ w / x ]_ A )
15	14	eqeq2d 2182	. . . . . . 7 |- ( y = w -> ( A = B <-> A = [_ w / x ]_ A ) )
16	8, 15	sbciegf 2986	. . . . . 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 3058	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
19	18	eqeq1d 2179	. . . . 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 2986	. . 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 1348   e. wcel 2141   _Vcvv 2730   [.wsbc 2955   [_csb 3049   class class class wbr 3989   {copab 4049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051

This theorem is referenced by:  eqer  6545