Theorem csbabgOLD 2589

Description: Move substitution into a class abstraction.

Assertion

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

Distinct variable groups:   y,A   x,y

Proof of Theorem csbabgOLD

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		visset 2295	. . . . . . 7 |- z e. _V
3		sbccomg 2526	. . . . . . 7 |- ((z e. _V /\ A e. _V) -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
4	2, 3	mpan 759	. . . . . 6 |- (A e. _V -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
5		df-clab 1872	. . . . . . 7 |- (z e. {y | ph} <-> [z / y]ph)
6	5	sbcbii 2506	. . . . . 6 |- (A e. _V -> ([A / x]z e. {y | ph} <-> [A / x][z / y]ph))
7	4, 6	bitr4d 590	. . . . 5 |- (A e. _V -> ([z / y][A / x]ph <-> [A / x]z e. {y | ph}))
8	7	abbidv 2008	. . . 4 |- (A e. _V -> {z | [z / y][A / x]ph} = {z | [A / x]z e. {y | ph}})
9		ax-17 1317	. . . . 5 |- ([A / x]ph -> A.z[A / x]ph)
10		hbs1 1722	. . . . 5 |- ([z / y][A / x]ph -> A.y[z / y][A / x]ph)
11		sbequ12 1545	. . . . 5 |- (y = z -> ([A / x]ph <-> [z / y][A / x]ph))
12	9, 10, 11	cbvab 2419	. . . 4 |- {y | [A / x]ph} = {z | [z / y][A / x]ph}
13	8, 12	syl5eq 1940	. . 3 |- (A e. _V -> {y | [A / x]ph} = {z | [A / x]z e. {y | ph}})
14		df-csb 2541	. . 3 |- [_A / x]_{y | ph} = {z | [A / x]z e. {y | ph}}
15	13, 14	syl6reqr 1947	. 2 |- (A e. _V -> [_A / x]_{y | ph} = {y | [A / x]ph})
16	1, 15	syl 12	1 |- (A e. B -> [_A / x]_{y | ph} = {y | [A / x]ph})

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

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

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