Theorem sbex 2023

Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)

Assertion

Ref	Expression
sbex	|- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Distinct variable groups:   x, y   x, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sbex

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbexyz 2022	. . . 4 |- ( [ w / y ] E. x ph <-> E. x [ w / y ] ph )
2	1	sbbii 1779	. . 3 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / w ] E. x [ w / y ] ph )
3		sbexyz 2022	. . 3 |- ( [ z / w ] E. x [ w / y ] ph <-> E. x [ z / w ] [ w / y ] ph )
4	2, 3	bitri 184	. 2 |- ( [ z / w ] [ w / y ] E. x ph <-> E. x [ z / w ] [ w / y ] ph )
5		ax-17 1540	. . 3 |- ( E. x ph -> A. w E. x ph )
6	5	sbco2vh 1964	. 2 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / y ] E. x ph )
7		ax-17 1540	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2vh 1964	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	exbii 1619	. 2 |- ( E. x [ z / w ] [ w / y ] ph <-> E. x [ z / y ] ph )
10	4, 6, 9	3bitr3i 210	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   <-> wb 105   E.wex 1506   [wsb 1776

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-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  sbabel  2366  sbcex2  3043  sbcexg  3044