Theorem sbex 1923

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 1922	. . . 4 |- ( [ w / y ] E. x ph <-> E. x [ w / y ] ph )
2	1	sbbii 1690	. . 3 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / w ] E. x [ w / y ] ph )
3		sbexyz 1922	. . 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 1460	. . 3 |- ( E. x ph -> A. w E. x ph )
6	5	sbco2v 1864	. 2 |- ( [ z / w ] [ w / y ] E. x ph <-> [ z / y ] E. x ph )
7		ax-17 1460	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2v 1864	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	exbii 1537	. 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 1422   [wsb 1687

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688

This theorem is referenced by:  sbabel  2248  sbcex2  2878  sbcexg  2879