Theorem eqerlem 5328

Description: Lemma for eqer 5329.

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 3566	. 2 |- (zRw <-> [w / y][z / x]A = B)
3		visset 2295	. . . . 5 |- z e. _V
4		sbceq1dig 2557	. . . . 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 2295	. . . . . 6 |- y e. _V
7		ax-17 1317	. . . . . 6 |- (v e. B -> A.x v e. B)
8		eqer.1	. . . . . 6 |- (x = y -> A = B)
9	6, 7, 8	csbief 2576	. . . . 5 |- [_y / x]_A = B
10	9	eqeq2i 1894	. . . 4 |- ([_z / x]_A = [_y / x]_A <-> [_z / x]_A = B)
11	5, 10	bitr4i 193	. . 3 |- ([z / x]A = B <-> [_z / x]_A = [_y / x]_A)
12	11	sbbii 1538	. 2 |- ([w / y][z / x]A = B <-> [w / y][_z / x]_A = [_y / x]_A)
13		visset 2295	. . . 4 |- w e. _V
14		sbceq2dig 2559	. . . 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 2547	. . . . 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 1894	. . 3 |- ([_z / x]_A = [_w / y]_[_y / x]_A <-> [_z / x]_A = [_w / x]_A)
19	15, 18	bitri 190	. 2 |- ([w / y][_z / x]_A = [_y / x]_A <-> [_z / x]_A = [_w / x]_A)
20	2, 12, 19	3bitri 194	1 |- (zRw <-> [_z / x]_A = [_w / x]_A)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  _Vcvv 2292  [_csb 2540   class class class wbr 3338  {copab 3395

This theorem is referenced by:  eqer 5329

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396

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