Theorem spi 2221

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1828. Contrary to the rule of generalization, its closed form is valid, see sp 2220. (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 2220	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

This proof depends on syntax axioms:  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  dariiALT  2691  barbariALT  2695  festinoALT  2700  barocoALT  2702  daraptiALT  2710  nfnfc  2935  axac2  10537  axac  10538  axaci  10539  bnj864  35545  axsepg2  35791  axsepg4  35794  axpowg2  35798  axpowg3  35799  in-ax8  36993  bj-snexg  37927  bj-unexg  37931  bj-adjg1  37936  sticksstones1  43176  sticksstones2  43177  rr-grothprim  45269  rr-grothshort  45273