Theorem csbopabg 3409

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 2299	. 2 |- (A e. B -> A e. _V)
2		df-opab 3396	. . . 4 |- {<.y, z>. | ph} = {w | E.yE.z(w = <.y, z>. /\ ph)}
3	2	csbeq2i 2563	. . 3 |- (A e. _V -> [_A / x]_{<.y, z>. | ph} = [_A / x]_{w | E.yE.z(w = <.y, z>. /\ ph)})
4		csbabg 2588	. . . 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 2501	. . . . . 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 2501	. . . . . . . 8 |- (A e. _V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z[A / x](w = <.y, z>. /\ ph)))
7		sbcang 2497	. . . . . . . . . 10 |- (A e. _V -> ([A / x](w = <.y, z>. /\ ph) <-> ([A / x]w = <.y, z>. /\ [A / x]ph)))
8		ax-17 1317	. . . . . . . . . . . 12 |- (w = <.y, z>. -> A.x w = <.y, z>.)
9	8	sbcgf 2520	. . . . . . . . . . 11 |- (A e. _V -> ([A / x]w = <.y, z>. <-> w = <.y, z>.))
10	9	anbi1d 679	. . . . . . . . . 10 |- (A e. _V -> (([A / x]w = <.y, z>. /\ [A / x]ph) <-> (w = <.y, z>. /\ [A / x]ph)))
11	7, 10	bitrd 587	. . . . . . . . 9 |- (A e. _V -> ([A / x](w = <.y, z>. /\ ph) <-> (w = <.y, z>. /\ [A / x]ph)))
12	11	exbidv 1657	. . . . . . . 8 |- (A e. _V -> (E.z[A / x](w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
13	6, 12	bitrd 587	. . . . . . 7 |- (A e. _V -> ([A / x]E.z(w = <.y, z>. /\ ph) <-> E.z(w = <.y, z>. /\ [A / x]ph)))
14	13	exbidv 1657	. . . . . 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 587	. . . . 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 2008	. . . 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 1925	. . 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 3396	. . . . 5 |- {<.y, z>. | [A / x]ph} = {w | E.yE.z(w = <.y, z>. /\ [A / x]ph)}
19	18	eqcomi 1888	. . . 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 1929	. 2 |- (A e. _V -> [_A / x]_{<.y, z>. | ph} = {<.y, z>. | [A / x]ph})
22	1, 21	syl 12	1 |- (A e. B -> [_A / x]_{<.y, z>. | ph} = {<.y, z>. | [A / x]ph})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540  <.cop 3046  {copab 3395

This theorem is referenced by:  fsumcnlem 9267

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-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

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-v 2294  df-sbc 2454  df-csb 2541  df-opab 3396

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