Theorem sbccsb2g 3123

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 2193	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	sbcbii 3058	. 2 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
3		sbcel12g 3108	. . 3 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } ) )
4		csbvarg 3121	. . . 4 |- ( A e. V -> [_ A / x ]_ x = A )
5	4	eleq1d 2274	. . 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 2176   {cab 2191   [.wsbc 2998   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094

This theorem is referenced by: (None)