Theorem spi 2177

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1798. Contrary to the rule of generalization, its closed form is valid, see sp 2176. (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 2176	. 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 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  dariiALT  2667  barbariALT  2671  festinoALT  2676  barocoALT  2678  daraptiALT  2686  nfnfc  2919  kmlem2  9907  axac2  10222  axac  10223  axaci  10224  bnj864  32902  sticksstones1  40102  sticksstones2  40103  rr-grothprim  41918  rr-grothshort  41922