Theorem csbabg 2588

Description: Move substitution into a class abstraction. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Distinct variable groups:   y,A   x,y

Proof of Theorem csbabg

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- z e. _V
2		sbccomg 2526	. . . . 5 |- ((z e. _V /\ A e. B) -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
3	1, 2	mpan 759	. . . 4 |- (A e. B -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
4		df-clab 1872	. . . 4 |- (z e. {y | [A / x]ph} <-> [z / y][A / x]ph)
5	3, 4	syl5bb 591	. . 3 |- (A e. B -> (z e. {y | [A / x]ph} <-> [A / x][z / y]ph))
6		df-clab 1872	. . . 4 |- (z e. {y | ph} <-> [z / y]ph)
7	6	sbcbii 2506	. . 3 |- (A e. B -> ([A / x]z e. {y | ph} <-> [A / x][z / y]ph))
8		sbcel2g 2558	. . 3 |- (A e. B -> ([A / x]z e. {y | ph} <-> z e. [_A / x]_{y | ph}))
9	5, 7, 8	3bitr2rd 606	. 2 |- (A e. B -> (z e. [_A / x]_{y | ph} <-> z e. {y | [A / x]ph}))
10	9	eqrdv 1882	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 is referenced by:  csbopabg 3409

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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