Theorem wl-equsb4

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019)

	Ref	Expression
Assertion	wl-equsb4	|- ( -. A. x x = z -> ( [ y / x ] y = z <-> y = z ) )

Proof

Step	Hyp	Ref	Expression
1		nfeqf	|- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x y = z )
2	1	ex	|- ( -. A. x x = y -> ( -. A. x x = z -> F/ x y = z ) )
3		sbft	|- ( F/ x y = z -> ( [ y / x ] y = z <-> y = z ) )
4	2 3	syl6com	|- ( -. A. x x = z -> ( -. A. x x = y -> ( [ y / x ] y = z <-> y = z ) ) )
5		sbequ12r	|- ( y = x -> ( [ y / x ] y = z <-> y = z ) )
6	5	equcoms	|- ( x = y -> ( [ y / x ] y = z <-> y = z ) )
7	6	sps	|- ( A. x x = y -> ( [ y / x ] y = z <-> y = z ) )
8	4 7	pm2.61d2	|- ( -. A. x x = z -> ( [ y / x ] y = z <-> y = z ) )