Theorem sbccsb2g 3102

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 2177	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	sbcbii 3037	. 2 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
3		sbcel12g 3087	. . 3 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } ) )
4		csbvarg 3100	. . . 4 |- ( A e. V -> [_ A / x ]_ x = A )
5	4	eleq1d 2258	. . 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 2160   {cab 2175   [.wsbc 2977   [_csb 3072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073

This theorem is referenced by: (None)