Theorem nf5-1 2023

Description: One direction of nf5 2116 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)

Assertion

Ref	Expression
nf5-1	|- ( A. x ( ph -> A. x ph ) -> F/ x ph )

Proof of Theorem nf5-1

Step	Hyp	Ref	Expression
1		exim 1761	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> E. x A. x ph ) )
2		hbe1a 2022	. . 3 |- ( E. x A. x ph -> A. x ph )
3	1, 2	syl6 35	. 2 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> A. x ph ) )
4	3	nfd 1716	1 |- ( A. x ( ph -> A. x ph ) -> F/ x ph )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   F/wnf 1708

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-10 2019

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  nf5i  2024  nf5dv  2025  nf5dh  2026  nf5d  2118  hbnt  2144