Theorem cbvrexsv 2695

Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
cbvrexsv	|- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )

Distinct variable groups:   x, A   ph, y   y, A

Allowed substitution hint:   ph( x)

Proof of Theorem cbvrexsv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1508	. . 3 |- F/ z ph
2		nfs1v 1919	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1751	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvrex 2677	. 2 |- ( E. x e. A ph <-> E. z e. A [ z / x ] ph )
5		nfv 1508	. . . 4 |- F/ y ph
6	5	nfsb 1926	. . 3 |- F/ y [ z / x ] ph
7		nfv 1508	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1820	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvrex 2677	. 2 |- ( E. z e. A [ z / x ] ph <-> E. y e. A [ y / x ] ph )
10	4, 9	bitri 183	1 |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 104   [wsb 1742   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441

This theorem is referenced by:  rspesbca  3021  rexxpf  4732