Theorem spi 2220

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1825. Contrary to the rule of generalization, its closed form is valid, see sp 2219. (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 2219	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  dariiALT  2693  barbariALT  2697  festinoALT  2702  barocoALT  2704  daraptiALT  2712  nfnfc  2937  axac2  10451  axac  10452  axaci  10453  bnj864  35291  axsepg2  35534  axsepg4  35537  axpowg2  35541  axpowg3  35542  in-ax8  36717  bj-snexg  37651  bj-unexg  37655  bj-adjg1  37660  sticksstones1  42894  sticksstones2  42895  rr-grothprim  44993  rr-grothshort  44997