Theorem spi 2185

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1795. 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 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  dariiALT  2660  barbariALT  2664  festinoALT  2669  barocoALT  2671  daraptiALT  2679  nfnfc  2905  kmlem2  10111  axac2  10425  axac  10426  axaci  10427  bnj864  34918  in-ax8  36207  bj-snexg  37017  bj-unexg  37021  bj-adjg1  37026  sticksstones1  42129  sticksstones2  42130  rr-grothprim  44282  rr-grothshort  44286