Theorem cbvralsv 3099

Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

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

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

Allowed substitution hint:   ph( x)

Proof of Theorem cbvralsv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1683	. . 3 |- F/ z ph
2		nfs1v 2164	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1961	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 3084	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1683	. . . 4 |- F/ y ph
6	5	nfsb 2168	. . 3 |- F/ y [ z / x ] ph
7		nfv 1683	. . 3 |- F/ z [ y / x ] ph
8		sbequ 2090	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 3084	. 2 |- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4, 9	bitri 249	1 |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 184   [wsb 1711   A.wral 2814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819

This theorem is referenced by:  sbralie  3101  rspsbc  3421  ralxpf  5147  tfinds  6672  tfindes  6675