Theorem spi 2218

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1814. Contrary to the rule of generalization, its closed form is valid, see sp 2217. (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 2217	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  dariiALT  2691  barbariALT  2695  festinoALT  2700  barocoALT  2702  daraptiALT  2710  nfnfc  2935  axac2  10417  axac  10418  axaci  10419  bnj864  35178  axsepg2  35397  axsepg4  35400  axpowg2  35404  axpowg3  35405  in-ax8  36545  bj-snexg  37480  bj-unexg  37484  bj-adjg1  37489  sticksstones1  42724  sticksstones2  42725  rr-grothprim  44837  rr-grothshort  44841