Theorem spi 2178

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1798. Contrary to the rule of generalization, its closed form is valid, see sp 2177. (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 2177	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  dariiALT  2662  barbariALT  2666  festinoALT  2671  barocoALT  2673  daraptiALT  2681  nfnfc  2916  kmlem2  10146  axac2  10461  axac  10462  axaci  10463  bnj864  33933  bj-snexg  35915  bj-unexg  35919  bj-adjg1  35924  sticksstones1  40962  sticksstones2  40963  rr-grothprim  43059  rr-grothshort  43063