Theorem sbccsbg 3074

Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem sbccsbg

Step	Hyp	Ref	Expression
1		abid 2153	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3010	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2g 3066	. 2 |- ( A e. V -> ( [. A / x ]. y e. { y | ph } <-> y e. [_ A / x ]_ { y | ph } ) )
4	2, 3	bitr3id 193	1 |- ( A e. V -> ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   {cab 2151   [.wsbc 2951   [_csb 3045

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by: (None)