Theorem sbcexg 3017

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Assertion

Ref	Expression
sbcexg	|- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )

Distinct variable groups:   x, A   x, y

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

Proof of Theorem sbcexg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2965	. 2 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
2		dfsbcq2 2965	. . 3 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
3	2	exbidv 1825	. 2 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
4		sbex 2004	. 2 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
5	1, 3, 4	vtoclbg 2798	1 |- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   [wsb 1762   e. wcel 2148   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963

This theorem is referenced by:  sbcabel  3044  csbunig  3817  csbxpg  4706  csbrng  5088