Theorem spi 2172

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1789. Contrary to the rule of generalization, its closed form is valid, see sp 2171. (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 2171	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-ex 1774

This theorem is referenced by:  dariiALT  2654  barbariALT  2658  festinoALT  2663  barocoALT  2665  daraptiALT  2673  nfnfc  2905  kmlem2  10174  axac2  10489  axac  10490  axaci  10491  bnj864  34640  bj-snexg  36600  bj-unexg  36604  bj-adjg1  36609  sticksstones1  41704  sticksstones2  41705  rr-grothprim  43819  rr-grothshort  43823