Theorem sbcex2 2957

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 2912	. 2 |- ( [. A / y ]. E. x ph -> A e. _V )
2		sbcex 2912	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	exlimiv 1577	. 2 |- ( E. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2907	. . 3 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5		dfsbcq2 2907	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	exbidv 1797	. . 3 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7		sbex 1977	. . 3 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2742	. 2 |- ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 693	1 |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Syntax hints:   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   [wsb 1735   _Vcvv 2681   [.wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  csbdmg  4728