Theorem sbcex2 3043

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

Assertion

Ref	Expression
sbcex2	|- ( [. 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)

Proof of Theorem sbcex2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2998	. 2 |- ( [. A / y ]. E. x ph -> A e. _V )
2		sbcex 2998	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	exlimiv 1612	. 2 |- ( E. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2992	. . 3 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5		dfsbcq2 2992	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	exbidv 1839	. . 3 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7		sbex 2023	. . 3 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2825	. 2 |- ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 705	1 |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Syntax hints:   <-> wb 105   = wceq 1364   E.wex 1506   [wsb 1776   e. wcel 2167   _Vcvv 2763   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  csbdmg  4860