Theorem spi 2180

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1790. Contrary to the rule of generalization, its closed form is valid, see sp 2179. (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 2179	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-ex 1775

This theorem is referenced by:  dariiALT  2662  barbariALT  2666  festinoALT  2671  barocoALT  2673  daraptiALT  2681  nfnfc  2914  kmlem2  10184  axac2  10498  axac  10499  axaci  10500  bnj864  34876  in-ax8  36167  bj-snexg  36977  bj-unexg  36981  bj-adjg1  36986  sticksstones1  42089  sticksstones2  42090  rr-grothprim  44262  rr-grothshort  44266