Theorem spimw 1926

Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Hypotheses

Ref	Expression
spimw.1	|- ( -. ps -> A. x -. ps )
spimw.2	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
spimw	|- ( A. x ph -> ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem spimw

Step	Hyp	Ref	Expression
1		ax6v 1889	. 2 |- -. A. x -. x = y
2		spimw.1	. . 3 |- ( -. ps -> A. x -. ps )
3		spimw.2	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	spimfw 1878	. 2 |- ( -. A. x -. x = y -> ( A. x ph -> ps ) )
5	1, 4	ax-mp 5	1 |- ( A. x ph -> ps )

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:  spimvw  1927  spnfw  1928  cbvaliw  1933  spfw  1965  spfwOLD  1966