Theorem sbcexg 2893

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 2843	. 2 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
2		dfsbcq2 2843	. . 3 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
3	2	exbidv 1753	. 2 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
4		sbex 1928	. 2 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
5	1, 3, 4	vtoclbg 2680	1 |- ( A e. V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438   [wsb 1692   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841

This theorem is referenced by:  sbcabel  2920  csbunig  3661  csbxpg  4519  csbrng  4892