Theorem sbex 2128

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

Assertion

Ref	Expression
sbex	|- ([z / y]E.xph <-> 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 2062	. . 3 |- ([z / y] -. A.x -. ph <-> -. [z / y]A.x -. ph)
2		sbal 2127	. . . 4 |- ([z / y]A.x -. ph <-> A.x[z / y] -. ph)
3		sbn 2062	. . . . 5 |- ([z / y] -. ph <-> -. [z / y]ph)
4	3	albii 1566	. . . 4 |- (A.x[z / y] -. ph <-> A.x -. [z / y]ph)
5	2, 4	bitri 240	. . 3 |- ([z / y]A.x -. ph <-> A.x -. [z / y]ph)
6	1, 5	xchbinx 301	. 2 |- ([z / y] -. A.x -. ph <-> -. A.x -. [z / y]ph)
7		df-ex 1542	. . 3 |- (E.xph <-> -. A.x -. ph)
8	7	sbbii 1653	. 2 |- ([z / y]E.xph <-> [z / y] -. A.x -. ph)
9		df-ex 1542	. 2 |- (E.x[z / y]ph <-> -. A.x -. [z / y]ph)
10	6, 8, 9	3bitr4i 268	1 |- ([z / y]E.xph <-> E.x[z / y]ph)

Syntax hints:  -. wn 3   <-> wb 176  A.wal 1540  E.wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbmo  2234  sbabel  2516  sbcex2  3096  sbcexg  3097