Theorem spi 2185

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1793. Contrary to the rule of generalization, its closed form is valid, see sp 2184. (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 2184	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778

This theorem is referenced by:  dariiALT  2669  barbariALT  2673  festinoALT  2678  barocoALT  2680  daraptiALT  2688  nfnfc  2921  kmlem2  10215  axac2  10529  axac  10530  axaci  10531  bnj864  34890  in-ax8  36182  bj-snexg  36992  bj-unexg  36996  bj-adjg1  37001  sticksstones1  42095  sticksstones2  42096  rr-grothprim  44264  rr-grothshort  44268