Theorem spi 2179

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1799. Contrary to the rule of generalization, its closed form is valid, see sp 2178. (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 2178	. 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 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  dariiALT  2667  barbariALT  2671  festinoALT  2676  barocoALT  2678  daraptiALT  2686  nfnfc  2918  kmlem2  9838  axac2  10153  axac  10154  axaci  10155  bnj864  32802  sticksstones1  40030  sticksstones2  40031  rr-grothprim  41807  rr-grothshort  41811