Theorem csbopabg 2683

Description: Move substitution into a class abstraction.

Assertion

Ref	Expression
csbopabg	|- (A e. B -> [_A / x]_{<.y, z>. | ph} = {<.y, z>. | [A / x]ph})

Distinct variable groups:   y,z,A   x,y,z

Proof of Theorem csbopabg

Step	Hyp	Ref	Expression
1		elisset 1820	. 2 |- (A e. B -> A e. V)
2		df-opab 2672	. . . 4 |- {<.y, z>. | ph} = {w | E.yE.z(w = <.y, z>. /\ ph)}
3	2	csbeq2i 2023	. . 3 |- (A e. V -> [_A / x]_{<.y, z>. | ph} = [_A / x]_{w | E.yE.z(w = <.y, z>. /\ ph)})
4		csbabg 2046	. . . 4 |- (A e. V -> [_A / x]_{w | E.yE.z(w = <.y, z>. /\ ph)} = {w | [A / x]E.yE.z(w = <.y, z>. /\ ph)})
5		sbcexg 1978	. . . . . 6 |- (A e. V -> ([A / x]E.yE.z(w = <.y, z>. /\ ph) <-> E.y[A / x]E.z(w = <.y, z>. /\ ph)))
6		sbcexg 1978	. . . . . . . 8 |- (A e. V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z[A / x](w = <.y, z>. /\ ph)))
7		sbcang 1974	. . . . . . . . . 10 |- (A e. V -> ([A / x](w = <.y, z>. /\ ph) <-> ([A / x]w = <.y, z>. /\ [A / x]ph)))
8		ax-17 973	. . . . . . . . . . . 12 |- (w = <.y, z>. -> A.x w = <.y, z>.)
9	8	sbcgf 1989	. . . . . . . . . . 11 |- (A e. V -> ([A / x]w = <.y, z>. <-> w = <.y, z>.))
10	9	anbi1d 619	. . . . . . . . . 10 |- (A e. V -> (([A / x]w = <.y, z>. /\ [A / x]ph) <-> (w = <.y, z>. /\ [A / x]ph)))
11	7, 10	bitrd 530	. . . . . . . . 9 |- (A e. V -> ([A / x](w = <.y, z>. /\ ph) <-> (w = <.y, z>. /\ [A / x]ph)))
12	11	exbidv 1281	. . . . . . . 8 |- (A e. V -> (E.z[A / x](w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
13	6, 12	bitrd 530	. . . . . . 7 |- (A e. V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
14	13	exbidv 1281	. . . . . 6 |- (A e. V -> (E.y[A / x]E.z(w = <.y, z>. /\ ph) <-> E.yE.z(w = <.y, z>. /\ [A / x]ph)))
15	5, 14	bitrd 530	. . . . 5 |- (A e. V -> ([A / x]E.yE.z(w = <.y, z>. /\ ph) <-> E.yE.z(w = <.y, z>. /\ [A / x]ph)))
16	15	abbidv 1580	. . . 4 |- (A e. V -> {w | [A / x]E.yE.z(w = <.y, z>. /\ ph)} = {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)})
17	4, 16	eqtrd 1510	. . 3 |- (A e. V -> [_A / x]_{w | E.yE.z(w = <.y, z>. /\ ph)} = {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)})
18		df-opab 2672	. . . . 5 |- {<.y, z>. | [A / x]ph} = {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)}
19	18	eqcomi 1482	. . . 4 |- {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)} = {<.y, z>. | [A / x]ph}
20	19	a1i 8	. . 3 |- (A e. V -> {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)} = {<.y, z>. | [A / x]ph})
21	3, 17, 20	3eqtrd 1514	. 2 |- (A e. V -> [_A / x]_{<.y, z>. | ph} = {<.y, z>. | [A / x]ph})
22	1, 21	syl 10	1 |- (A e. B -> [_A / x]_{<.y, z>. | ph} = {<.y, z>. | [A / x]ph})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  {cab 1466  Vcvv 1814  [_csb 2004  <.cop 2415  {copab 2671

This theorem is referenced by:  fsumcnlem 7986

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005  df-opab 2672

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