Theorem spi 2176

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1796. Contrary to the rule of generalization, its closed form is valid, see sp 2175. (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 2175	. 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 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by:  dariiALT  2665  barbariALT  2669  festinoALT  2674  barocoALT  2676  daraptiALT  2684  nfnfc  2916  kmlem2  10000  axac2  10315  axac  10316  axaci  10317  bnj864  33142  bj-snexg  35311  bj-unexg  35315  sticksstones1  40352  sticksstones2  40353  rr-grothprim  42228  rr-grothshort  42232