Theorem sb9v 1954

Description: Like sb9 1955 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 1912	. 2 |- ( [ x / y ] ph -> A. y [ x / y ] ph )
2		hbs1 1912	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1745	. . . 4 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
4	3	equcoms 1685	. . 3 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
5		sbequ12 1745	. . 3 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	bitr3d 189	. 2 |- ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
7	1, 2, 6	cbvalh 1727	1 |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 104   A.wal 1330   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sb9  1955