Theorem spfalw 1929

Description: Version of sp 2053 when ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)

Hypothesis

Ref	Expression
spfalw.1	|- -. ph

Assertion

Ref	Expression
spfalw	|- ( A. x ph -> ph )

Proof of Theorem spfalw

Step	Hyp	Ref	Expression
1		spfalw.1	. . 3 |- -. ph
2	1	hbth 1729	. 2 |- ( -. ph -> A. x -. ph )
3	2	spnfw 1928	1 |- ( A. x ph -> ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

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-6 1888

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

This theorem is referenced by:  ax6dgen  2005