Theorem sbex 1921

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 1920	. . . 4 |- ( [ w / y ] E. x ph <-> E. x [ w / y ] ph )
2	1	sbbii 1688	. . 3 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / w ] E. x [ w / y ] ph )
3		sbexyz 1920	. . 3 |- ( [ z / w ] E. x [ w / y ] ph <-> E. x [ z / w ] [ w / y ] ph )
4	2, 3	bitri 182	. 2 |- ( [ z / w ] [ w / y ] E. x ph <-> E. x [ z / w ] [ w / y ] ph )
5		ax-17 1459	. . 3 |- ( E. x ph -> A. w E. x ph )
6	5	sbco2v 1862	. 2 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / y ] E. x ph )
7		ax-17 1459	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2v 1862	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	exbii 1536	. 2 |- ( E. x [ z / w ] [ w / y ] ph <-> E. x [ z / y ] ph )
10	4, 6, 9	3bitr3i 208	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   <-> wb 103   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbabel  2244  sbcex2  2867  sbcexg  2868