Theorem spi 2185

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 2184. (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 2184	. 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 2008  ax-12 2178

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

This theorem is referenced by:  dariiALT  2659  barbariALT  2663  festinoALT  2668  barocoALT  2670  daraptiALT  2678  nfnfc  2904  kmlem2  10105  axac2  10419  axac  10420  axaci  10421  bnj864  34912  in-ax8  36212  bj-snexg  37022  bj-unexg  37026  bj-adjg1  37031  sticksstones1  42134  sticksstones2  42135  rr-grothprim  44289  rr-grothshort  44293