Theorem sbcex2 2812

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 2772	. 2 |- ( [. A / y ]. E. x ph -> A e. _V )
2		sbcex 2772	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	exlimiv 1489	. 2 |- ( E. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2767	. . 3 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5		dfsbcq2 2767	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	exbidv 1706	. . 3 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7		sbex 1880	. . 3 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2614	. 2 |- ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 620	1 |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Syntax hints:   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   [wsb 1645   _Vcvv 2557   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  csbdmg  4529