Theorem sbccsbg 3153

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 2217	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3088	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2g 3145	. 2 |- ( A e. V -> ( [. A / x ]. y e. { y | ph } <-> y e. [_ A / x ]_ { y | ph } ) )
4	2, 3	bitr3id 194	1 |- ( A e. V -> ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   {cab 2215   [.wsbc 3028   [_csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by: (None)