Theorem spim 2254

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2254 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-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	. 2 |- F/ x ps
2		ax6e 2250	. . 3 |- E. x x = y
3		spim.2	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	eximii 1764	. 2 |- E. x ( ph -> ps )
5	1, 4	19.36i 2099	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708

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-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  spimvALT  2258  chvar  2262  cbv3  2265  setrec2fun  42439