Theorem sb9v 2031

Description: Like sb9 2032 but with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

Ref	Expression
sb9v	|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb9v

Step	Hyp	Ref	Expression
1		hbs1 1991	. 2 |- ( [ x / y ] ph -> A. y [ x / y ] ph )
2		hbs1 1991	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1819	. . . 4 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
4	3	equcoms 1756	. . 3 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
5		sbequ12 1819	. . 3 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	bitr3d 190	. 2 |- ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
7	1, 2, 6	cbvalh 1801	1 |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 105   A.wal 1395   [wsb 1810

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sb9  2032