Theorem sbccsb2g 3079

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 2158	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	sbcbii 3014	. 2 |- ( [. A / x ]. x e. { x | ph } <-> [. A / x ]. ph )
3		sbcel12g 3064	. . 3 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> [_ A / x ]_ x e. [_ A / x ]_ { x | ph } ) )
4		csbvarg 3077	. . . 4 |- ( A e. V -> [_ A / x ]_ x = A )
5	4	eleq1d 2239	. . 3 |- ( A e. V -> ( [_ A / x ]_ x e. [_ A / x ]_ { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
6	3, 5	bitrd 187	. 2 |- ( A e. V -> ( [. A / x ]. x e. { x | ph } <-> A e. [_ A / x ]_ { x | ph } ) )
7	2, 6	bitr3id 193	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by: (None)