Theorem spi 2170

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 2169. (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 2169	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1532

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 1964  ax-7 2004  ax-12 2164

This theorem depends on definitions:  df-bi 206  df-ex 1775

This theorem is referenced by:  dariiALT  2656  barbariALT  2660  festinoALT  2665  barocoALT  2667  daraptiALT  2675  nfnfc  2910  kmlem2  10166  axac2  10481  axac  10482  axaci  10483  bnj864  34489  bj-snexg  36449  bj-unexg  36453  bj-adjg1  36458  sticksstones1  41550  sticksstones2  41551  rr-grothprim  43660  rr-grothshort  43664