Theorem sbcex2 3059

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 3014	. 2 |- ( [. A / y ]. E. x ph -> A e. _V )
2		sbcex 3014	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	exlimiv 1622	. 2 |- ( E. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 3008	. . 3 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5		dfsbcq2 3008	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	exbidv 1849	. . 3 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7		sbex 2033	. . 3 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2839	. 2 |- ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 706	1 |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Syntax hints:   <-> wb 105   = wceq 1373   E.wex 1516   [wsb 1786   e. wcel 2178   _Vcvv 2776   [.wsbc 3005

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006

This theorem is referenced by:  csbdmg  4891