Theorem cbvrexsv 2785

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 1577	. . 3 |- F/ z ph
2		nfs1v 1992	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1819	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvrex 2765	. 2 |- ( E. x e. A ph <-> E. z e. A [ z / x ] ph )
5		nfv 1577	. . . 4 |- F/ y ph
6	5	nfsb 1999	. . 3 |- F/ y [ z / x ] ph
7		nfv 1577	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1888	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvrex 2765	. 2 |- ( E. z e. A [ z / x ] ph <-> E. y e. A [ y / x ] ph )
10	4, 9	bitri 184	1 |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 105   [wsb 1810   E.wrex 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517

This theorem is referenced by:  rspesbca  3118  rexxpf  4883