Theorem wl-sbnf1

Description: Two ways expressing that x is effectively not free in ph . Simplified version of sbnf2 . Note: This theorem shows that sbnf2 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019)

	Ref	Expression
Assertion	wl-sbnf1	|- ( A. x F/ y ph -> ( F/ x ph <-> A. x A. y ( ph -> [ y / x ] ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		nf5	|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1	|- F/ x A. x F/ y ph
3		wl-sbhbt	|- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) )
4	2 3	albid	|- ( A. x F/ y ph -> ( A. x ( ph -> A. x ph ) <-> A. x A. y ( ph -> [ y / x ] ph ) ) )
5	1 4	syl5bb	|- ( A. x F/ y ph -> ( F/ x ph <-> A. x A. y ( ph -> [ y / x ] ph ) ) )