Theorem csbabg 3197

Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Distinct variable groups:   y,A   x,y

Allowed substitution hints:   ph(x,y)   A(x)   V(x,y)

Proof of Theorem csbabg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 3117	. . . 4 |- ( [.z / y ]. [.A / x ].ph <-> [.A / x ]. [.z / y ].ph)
2		df-clab 2340	. . . . 5 |- (z e. {y | [.A / x ].ph} <-> [z / y] [.A / x ].ph)
3		sbsbc 3050	. . . . 5 |- ([z / y] [.A / x ].ph <-> [.z / y ]. [.A / x ].ph)
4	2, 3	bitri 240	. . . 4 |- (z e. {y | [.A / x ].ph} <-> [.z / y ]. [.A / x ].ph)
5		df-clab 2340	. . . . . 6 |- (z e. {y | ph} <-> [z / y]ph)
6		sbsbc 3050	. . . . . 6 |- ([z / y]ph <-> [.z / y ].ph)
7	5, 6	bitri 240	. . . . 5 |- (z e. {y | ph} <-> [.z / y ].ph)
8	7	sbcbii 3101	. . . 4 |- ( [.A / x ].z e. {y | ph} <-> [.A / x ]. [.z / y ].ph)
9	1, 4, 8	3bitr4i 268	. . 3 |- (z e. {y | [.A / x ].ph} <-> [.A / x ].z e. {y | ph})
10		sbcel2g 3157	. . 3 |- (A e. V -> ( [.A / x ].z e. {y | ph} <-> z e. [_A / x]_{y | ph}))
11	9, 10	syl5rbb 249	. 2 |- (A e. V -> (z e. [_A / x]_{y | ph} <-> z e. {y | [.A / x ].ph}))
12	11	eqrdv 2351	1 |- (A e. V -> [_A / x]_{y | ph} = {y | [.A / x ].ph})

Syntax hints:   -> wi 4   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   [.wsbc 3046  [_csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbsng  3785  csbunig  3899  csbxpg  4813  csbrng  4966