Theorem sb9

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) Allow a shortening of sb9i . (Revised by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

	Ref	Expression
Assertion	sb9	|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sbequ12a	|- ( y = x -> ( [ x / y ] ph <-> [ y / x ] ph ) )
2	1	equcoms	|- ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
3	2	sps	|- ( A. x x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
4	3	dral1	|- ( A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
5		nfnae	|- F/ x -. A. x x = y
6		nfnae	|- F/ y -. A. x x = y
7		nfsb2	|- ( -. A. y y = x -> F/ y [ x / y ] ph )
8	7	naecoms	|- ( -. A. x x = y -> F/ y [ x / y ] ph )
9		nfsb2	|- ( -. A. x x = y -> F/ x [ y / x ] ph )
10	2	a1i	|- ( -. A. x x = y -> ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) ) )
11	5 6 8 9 10	cbv2	|- ( -. A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
12	4 11	pm2.61i	|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )