Theorem nf5-1 2024

Description: One direction of nf5 . (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 1599	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> E. x A. x ph ) )
2		hbe1a 2023	. . 3 |- ( E. x A. x ph -> A. x ph )
3	1, 2	syl6 33	. 2 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> A. x ph ) )
4	3	nfd2 2022	1 |- ( A. x ( ph -> A. x ph ) -> F/ x ph )

Syntax hints:   -> wi 4   A.wal 1351   F/wnf 1460   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  nf5d  2025