Theorem wl-sbnf1 31927

Description: Two ways expressing that x is effectively not free in ph. Simplified version of sbnf2 2278. Note: This theorem shows that sbnf2 2278 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)

Assertion

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

Proof of Theorem wl-sbnf1

Step	Hyp	Ref	Expression
1		df-nf 1678	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 1989	. . 3 |- F/ x A. x F/ y ph
3		wl-sbhbt 31926	. . 3 |- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) )
4	2, 3	albid 1973	. 2 |- ( A. x F/ y ph -> ( A. x ( ph -> A. x ph ) <-> A. x A. y ( ph -> [ y / x ] ph ) ) )
5	1, 4	syl5bb 265	1 |- ( A. x F/ y ph -> ( F/ x ph <-> A. x A. y ( ph -> [ y / x ] ph ) ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452   F/wnf 1677   [wsb 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808

This theorem is referenced by: (None)