Theorem wl-nfs1t

Description: If y is not free in ph , x is not free in [ y / x ] ph . Closed form of nfs1 . (Contributed by Wolf Lammen, 27-Jul-2019)

	Ref	Expression
Assertion	wl-nfs1t	|- ( F/ y ph -> F/ x [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	|- ( y = x -> ( [ y / x ] ph <-> ph ) )
2	1	equcoms	|- ( x = y -> ( [ y / x ] ph <-> ph ) )
3	2	sps	|- ( A. x x = y -> ( [ y / x ] ph <-> ph ) )
4	3	drnf1	|- ( A. x x = y -> ( F/ x [ y / x ] ph <-> F/ y ph ) )
5	4	biimprd	|- ( A. x x = y -> ( F/ y ph -> F/ x [ y / x ] ph ) )
6		nfsb2	|- ( -. A. x x = y -> F/ x [ y / x ] ph )
7	6	a1d	|- ( -. A. x x = y -> ( F/ y ph -> F/ x [ y / x ] ph ) )
8	5 7	pm2.61i	|- ( F/ y ph -> F/ x [ y / x ] ph )