Theorem spi 2177

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1797. 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 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1782

This theorem is referenced by:  dariiALT  2660  barbariALT  2664  festinoALT  2669  barocoALT  2671  daraptiALT  2679  nfnfc  2914  kmlem2  10096  axac2  10411  axac  10412  axaci  10413  bnj864  33623  bj-snexg  35578  bj-unexg  35582  bj-adjg1  35587  sticksstones1  40627  sticksstones2  40628  rr-grothprim  42702  rr-grothshort  42706