Theorem cbvralsv 2846

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 1619	. . 3 |- F/zph
2		nfs1v 2106	. . 3 |- F/x[z / x]ph
3		sbequ12 1919	. . 3 |- (x = z -> (ph <-> [z / x]ph))
4	1, 2, 3	cbvral 2831	. 2 |- (A.x e. A ph <-> A.z e. A [z / x]ph)
5		nfv 1619	. . . 4 |- F/yph
6	5	nfsb 2109	. . 3 |- F/y[z / x]ph
7		nfv 1619	. . 3 |- F/z[y / x]ph
8		sbequ 2060	. . 3 |- (z = y -> ([z / x]ph <-> [y / x]ph))
9	6, 7, 8	cbvral 2831	. 2 |- (A.z e. A [z / x]ph <-> A.y e. A [y / x]ph)
10	4, 9	bitri 240	1 |- (A.x e. A ph <-> A.y e. A [y / x]ph)

Syntax hints:   <-> wb 176  [wsb 1648  A.wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619

This theorem is referenced by:  sbralie  2848  rspsbc  3124  ralxpf  4827