Theorem spi 2196

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1802. Contrary to the rule of generalization, its closed form is valid, see sp 2195. (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 2195	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  dariiALT  2669  barbariALT  2673  festinoALT  2678  barocoALT  2680  daraptiALT  2688  nfnfc  2913  axac2  10379  axac  10380  axaci  10381  bnj864  35104  axsepg2  35321  axsepg4  35324  axpowg2  35328  axpowg3  35329  in-ax8  36452  bj-snexg  37387  bj-unexg  37391  bj-adjg1  37396  sticksstones1  42631  sticksstones2  42632  rr-grothprim  44744  rr-grothshort  44748