Theorem sbcexg 3032

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 2980	. 2 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
2		dfsbcq2 2980	. . 3 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
3	2	exbidv 1836	. 2 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
4		sbex 2016	. 2 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
5	1, 3, 4	vtoclbg 2813	1 |- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1503   [wsb 1773   e. wcel 2160   [.wsbc 2977

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by:  sbcabel  3059  csbunig  3832  csbxpg  4725  csbrng  5108