Theorem spi 2219

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1824. Contrary to the rule of generalization, its closed form is valid, see sp 2218. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
spi.1	⊢ ∀𝑥𝜑

Assertion

Ref	Expression
spi	⊢ 𝜑

Proof of Theorem spi

Step	Hyp	Ref	Expression
1		spi.1	. 2 ⊢ ∀𝑥𝜑
2		sp 2218	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

This proof depends on syntax axioms:  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  dariiALT  2692  barbariALT  2696  festinoALT  2701  barocoALT  2703  daraptiALT  2711  nfnfc  2936  axac2  10456  axac  10457  axaci  10458  bnj864  35319  axsepg2  35561  axsepg4  35564  axpowg2  35568  axpowg3  35569  in-ax8  36764  bj-snexg  37698  bj-unexg  37702  bj-adjg1  37707  sticksstones1  42941  sticksstones2  42942  rr-grothprim  45038  rr-grothshort  45042