Theorem spi 2192

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1797. Contrary to the rule of generalization, its closed form is valid, see sp 2191. (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 2191	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  dariiALT  2667  barbariALT  2671  festinoALT  2676  barocoALT  2678  daraptiALT  2686  nfnfc  2912  kmlem2  10074  axac2  10388  axac  10389  axaci  10390  bnj864  35097  in-ax8  36437  bj-snexg  37276  bj-unexg  37280  bj-adjg1  37285  sticksstones1  42510  sticksstones2  42511  rr-grothprim  44650  rr-grothshort  44654