Theorem cbvralsv 2754

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 1551	. . 3 |- F/ z ph
2		nfs1v 1967	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1794	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 2734	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1551	. . . 4 |- F/ y ph
6	5	nfsb 1974	. . 3 |- F/ y [ z / x ] ph
7		nfv 1551	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1863	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 2734	. 2 |- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4, 9	bitri 184	1 |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 105   [wsb 1785   A.wral 2484

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  sbralie  2756  rspsbc  3081  ralxpf  4824