Theorem sbcexg 3086

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 3034	. 2 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
2		dfsbcq2 3034	. . 3 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
3	2	exbidv 1873	. 2 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
4		sbex 2057	. 2 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
5	1, 3, 4	vtoclbg 2865	1 |- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   E.wex 1540   [wsb 1810   e. wcel 2202   [.wsbc 3031

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  sbcabel  3114  csbunig  3901  csbxpg  4807  csbrng  5198