Theorem sbccsbg 3126

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 2194	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3062	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2g 3118	. 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 2177   {cab 2192   [.wsbc 3002   [_csb 3097

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098

This theorem is referenced by: (None)