Theorem spim 1738

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1738 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)

Hypotheses

Ref	Expression
spim.1	|- F/ x ps
spim.2	|- ( x = y -> ( ph -> ps ) )

Assertion

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

Proof of Theorem spim

Step	Hyp	Ref	Expression
1		spim.1	. . 3 |- F/ x ps
2	1	nfri 1519	. 2 |- ( ps -> A. x ps )
3		spim.2	. 2 |- ( x = y -> ( ph -> ps ) )
4	2, 3	spimh 1737	1 |- ( A. x ph -> 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-i9 1530  ax-ial 1534

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

This theorem is referenced by:  cbv3  1742  chvar  1757  spimv  1811  2spim  14603