Theorem csbopabg 2678

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 1817	. 2 |- (A e. B -> A e. V)
2		df-opab 2667	. . . 4 |- {<.y, z>. | ph} = {w | E.yE.z(w = <.y, z>. /\ ph)}
3	2	csbeq2i 2020	. . 3 |- (A e. V -> [_A / x]_{<.y, z>. | ph} = [_A / x]_{w | E.yE.z(w = <.y, z>. /\ ph)})
4		csbabg 2043	. . . 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 1975	. . . . . 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 1975	. . . . . . . 8 |- (A e. V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z[A / x](w = <.y, z>. /\ ph)))
7		sbcang 1971	. . . . . . . . . 10 |- (A e. V -> ([A / x](w = <.y, z>. /\ ph) <-> ([A / x]w = <.y, z>. /\ [A / x]ph)))
8		ax-17 971	. . . . . . . . . . . 12 |- (w = <.y, z>. -> A.x w = <.y, z>.)
9	8	sbcgf 1986	. . . . . . . . . . 11 |- (A e. V -> ([A / x]w = <.y, z>. <-> w = <.y, z>.))
10	9	anbi1d 617	. . . . . . . . . 10 |- (A e. V -> (([A / x]w = <.y, z>. /\ [A / x]ph) <-> (w = <.y, z>. /\ [A / x]ph)))
11	7, 10	bitrd 528	. . . . . . . . 9 |- (A e. V -> ([A / x](w = <.y, z>. /\ ph) <-> (w = <.y, z>. /\ [A / x]ph)))
12	11	exbidv 1279	. . . . . . . 8 |- (A e. V -> (E.z[A / x](w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
13	6, 12	bitrd 528	. . . . . . 7 |- (A e. V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
14	13	exbidv 1279	. . . . . 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 528	. . . . 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 1577	. . . 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 1507	. . 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 2667	. . . . 5 |- {<.y, z>. | [A / x]ph} = {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)}
19	18	eqcomi 1479	. . . 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 1511	. 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 956   e. wcel 958  E.wex 980  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001  <.cop 2411  {copab 2666

This theorem is referenced by:  fsumcnlem 7989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002  df-opab 2667

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