Theorem sbccsb2g 3158

Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Assertion

Ref	Expression
sbccsb2g	|- ( A e. V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )

Proof of Theorem sbccsb2g

Step	Hyp	Ref	Expression
1		abid 2219	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	sbcbii 3092	. 2 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
3		sbcel12g 3143	. . 3 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } ) )
4		csbvarg 3156	. . . 4 |- ( A e. V -> [_ A / x ]_ x = A )
5	4	eleq1d 2300	. . 3 |- ( A e. V -> ( [_ A / x ]_ x e. [_ A / x ]_ { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
6	3, 5	bitrd 188	. 2 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
7	2, 6	bitr3id 194	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   {cab 2217   [.wsbc 3032   [_csb 3128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-csb 3129

This theorem is referenced by: (None)