Theorem cbvralsv 2997

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 1764	. . 3 |- F/ z ph
2		nfs1v 2266	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 2083	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 2982	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1764	. . . 4 |- F/ y ph
6	5	nfsb 2269	. . 3 |- F/ y [ z / x ] ph
7		nfv 1764	. . 3 |- F/ z [ y / x ] ph
8		sbequ 2205	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 2982	. 2 |- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4, 9	bitri 257	1 |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 189   [wsb 1800   A.wral 2736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1667  df-nf 1671  df-sb 1801  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741

This theorem is referenced by:  sbralie  2999  rspsbc  3313  ralxpf  4958  tfinds  6673  tfindes  6676  nn0min  28391