Theorem nf5d 2025

Description: Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nf5d.1	|- F/ x ph
nf5d.2	|- ( ph -> ( ps -> A. x ps ) )

Assertion

Ref	Expression
nf5d	|- ( ph -> F/ x ps )

Proof of Theorem nf5d

Step	Hyp	Ref	Expression
1		nf5d.1	. . 3 |- F/ x ph
2		nf5d.2	. . 3 |- ( ph -> ( ps -> A. x ps ) )
3	1, 2	alrimi 1522	. 2 |- ( ph -> A. x ( ps -> A. x ps ) )
4		nf5-1 2024	. 2 |- ( A. x ( ps -> A. x ps ) -> F/ x ps )
5	3, 4	syl 14	1 |- ( ph -> F/ x ps )

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

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:  nfabdw  2338