Theorem spi 2224

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1828. Contrary to the rule of generalization, its closed form is valid, see sp 2223. (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 2223	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

This proof depends on syntax axioms:  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  dariiALT  2693  barbariALT  2697  festinoALT  2702  barocoALT  2704  daraptiALT  2712  nfnfc  2938  axac2  10583  axac  10584  axaci  10585  bnj864  35663  axsepg2  35949  axsepg4  35952  axpowg2  35956  axpowg3  35957  in-ax8  37228  bj-snexg  38162  bj-unexg  38166  bj-adjg1  38171  sticksstones1  43462  sticksstones2  43463  rr-grothprim  45545  rr-grothshort  45549