Theorem sbccsbg 3036

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 2128	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 2972	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2g 3028	. 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 1481   {cab 2126   [.wsbc 2913   [_csb 3007

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by: (None)