Theorem spi 2226

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1822. Contrary to the rule of generalization, its closed form is valid, see sp 2225. (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 2225	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807

This theorem is referenced by:  dariiALT  2699  barbariALT  2703  festinoALT  2708  barocoALT  2710  daraptiALT  2718  nfnfc  2943  axac2  10446  axac  10447  axaci  10448  bnj864  35251  axsepg2  35472  axsepg4  35475  axpowg2  35479  axpowg3  35480  in-ax8  36621  bj-snexg  37554  bj-unexg  37558  bj-adjg1  37563  sticksstones1  42798  sticksstones2  42799  rr-grothprim  44895  rr-grothshort  44899