Theorem sbcex2 3486

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)

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 3445	. 2 |- ( [. A / y ]. E. x ph -> A e. _V )
2		sbcex 3445	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	exlimiv 1858	. 2 |- ( E. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 3438	. . 3 |- ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5		dfsbcq2 3438	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	exbidv 1850	. . 3 |- ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7		sbex 2463	. . 3 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 3267	. 2 |- ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 368	1 |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcabel  3517  csbuni  4466  csbxp  5200  csbdm  5318  sbcfung  5912  bnj89  30787  bnj985  31023  csbwrecsg  33173  csboprabg  33176  sbcexf  33918  onfrALTlem5  38757