Theorem sbex 2463

Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)

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

Step	Hyp	Ref	Expression
1		sbn 2391	. . 3 |- ( [ z / y ] -. A. x -. ph <-> -. [ z / y ] A. x -. ph )
2		sbal 2462	. . . 4 |- ( [ z / y ] A. x -. ph <-> A. x [ z / y ] -. ph )
3		sbn 2391	. . . . 5 |- ( [ z / y ] -. ph <-> -. [ z / y ] ph )
4	3	albii 1747	. . . 4 |- ( A. x [ z / y ] -. ph <-> A. x -. [ z / y ] ph )
5	2, 4	bitri 264	. . 3 |- ( [ z / y ] A. x -. ph <-> A. x -. [ z / y ] ph )
6	1, 5	xchbinx 324	. 2 |- ( [ z / y ] -. A. x -. ph <-> -. A. x -. [ z / y ] ph )
7		df-ex 1705	. . 3 |- ( E. x ph <-> -. A. x -. ph )
8	7	sbbii 1887	. 2 |- ( [ z / y ] E. x ph <-> [ z / y ] -. A. x -. ph )
9		df-ex 1705	. 2 |- ( E. x [ z / y ] ph <-> -. A. x -. [ z / y ] ph )
10	6, 8, 9	3bitr4i 292	1 |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   E.wex 1704   [wsb 1880

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

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

This theorem is referenced by:  sbmo  2515  sbabel  2793  sbcex2  3486  sbcexgOLD  38753