Theorem spi 2184

Description: Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1795. Contrary to the rule of generalization, its closed form is valid, see sp 2183. (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 2183	. 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  dariiALT  2665  barbariALT  2669  festinoALT  2674  barocoALT  2676  daraptiALT  2684  nfnfc  2917  kmlem2  10188  axac2  10502  axac  10503  axaci  10504  bnj864  34914  in-ax8  36203  bj-snexg  37013  bj-unexg  37017  bj-adjg1  37022  sticksstones1  42125  sticksstones2  42126  rr-grothprim  44297  rr-grothshort  44301