Theorem spi 2183

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1794. Contrary to the rule of generalization, its closed form is valid, see sp 2182. (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 2182	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by:  dariiALT  2664  barbariALT  2668  festinoALT  2673  barocoALT  2675  daraptiALT  2683  nfnfc  2910  kmlem2  10159  axac2  10473  axac  10474  axaci  10475  bnj864  34882  in-ax8  36171  bj-snexg  36981  bj-unexg  36985  bj-adjg1  36990  sticksstones1  42088  sticksstones2  42089  rr-grothprim  44257  rr-grothshort  44261