Theorem spi 2187

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1796. Contrary to the rule of generalization, its closed form is valid, see sp 2186. (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 2186	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  dariiALT  2661  barbariALT  2665  festinoALT  2670  barocoALT  2672  daraptiALT  2680  nfnfc  2907  kmlem2  10040  axac2  10354  axac  10355  axaci  10356  bnj864  34929  in-ax8  36257  bj-snexg  37067  bj-unexg  37071  bj-adjg1  37076  sticksstones1  42178  sticksstones2  42179  rr-grothprim  44332  rr-grothshort  44336