Theorem eqerlem 4267

Description: Lemma for eqer 4268.

Hypotheses

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

Assertion

Ref	Expression
eqerlem	|- (zRw <-> [_z / x]_A = [_w / x]_A)

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

Proof of Theorem eqerlem

Step	Hyp	Ref	Expression
1		eqer.2	. . 3 |- R = {<.x, y>. | A = B}
2	1	brabsb 2813	. 2 |- (zRw <-> [w / y][z / x]A = B)
3		visset 1811	. . . . 5 |- z e. V
4		sbceq1dig 2012	. . . . 5 |- (z e. V -> ([z / x]A = B <-> [_z / x]_A = B))
5	3, 4	ax-mp 7	. . . 4 |- ([z / x]A = B <-> [_z / x]_A = B)
6		visset 1811	. . . . . 6 |- y e. V
7		ax-17 970	. . . . . 6 |- (v e. B -> A.x v e. B)
8		eqer.1	. . . . . 6 |- (x = y -> A = B)
9	6, 7, 8	csbief 2030	. . . . 5 |- [_y / x]_A = B
10	9	eqeq2i 1484	. . . 4 |- ([_z / x]_A = [_y / x]_A <-> [_z / x]_A = B)
11	5, 10	bitr4 176	. . 3 |- ([z / x]A = B <-> [_z / x]_A = [_y / x]_A)
12	11	sbbii 1174	. 2 |- ([w / y][z / x]A = B <-> [w / y][_z / x]_A = [_y / x]_A)
13		visset 1811	. . . 4 |- w e. V
14		sbceq2dig 2014	. . . 4 |- (w e. V -> ([w / y][_z / x]_A = [_y / x]_A <-> [_z / x]_A = [_w / y]_[_y / x]_A))
15	13, 14	ax-mp 7	. . 3 |- ([w / y][_z / x]_A = [_y / x]_A <-> [_z / x]_A = [_w / y]_[_y / x]_A)
16		csbcog 2005	. . . . 5 |- (w e. V -> [_w / y]_[_y / x]_A = [_w / x]_A)
17	13, 16	ax-mp 7	. . . 4 |- [_w / y]_[_y / x]_A = [_w / x]_A
18	17	eqeq2i 1484	. . 3 |- ([_z / x]_A = [_w / y]_[_y / x]_A <-> [_z / x]_A = [_w / x]_A)
19	15, 18	bitr 173	. 2 |- ([w / y][_z / x]_A = [_y / x]_A <-> [_z / x]_A = [_w / x]_A)
20	2, 12, 19	3bitr 177	1 |- (zRw <-> [_z / x]_A = [_w / x]_A)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  [wsbc 1170  Vcvv 1809  [_csb 1999   class class class wbr 2616  {copab 2663

This theorem is referenced by:  eqer 4268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664

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