Theorem wl-sbnf1 33336

Description: Two ways expressing that x is effectively not free in ph. Simplified version of sbnf2 2439. Note: This theorem shows that sbnf2 2439 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		nf5 2116	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nfa1 2028	. . 3 |- F/ x A. x F/ y ph
3		wl-sbhbt 33335	. . 3 |- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) )
4	2, 3	albid 2090	. 2 |- ( A. x F/ y ph -> ( A. x ( ph -> A. x ph ) <-> A. x A. y ( ph -> [ y / x ] ph ) ) )
5	1, 4	syl5bb 272	1 |- ( A. x F/ y ph -> ( F/ x ph <-> A. x A. y ( ph -> [ y / x ] ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)