Theorem rspec 3229

Description: Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Hypothesis

Ref	Expression
rspec.1	⊢ ∀𝑥 ∈ 𝐴 𝜑

Assertion

Ref	Expression
rspec	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Proof of Theorem rspec

Step	Hyp	Ref	Expression
1		rspec.1	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝜑
2		rsp 3226	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-ral 3053

This theorem is referenced by:  rspec2  3257  vtoclri  3533  wfis  6312  wfis2f  6314  wfis2  6316  isarep2  6584  mpoexw  8026  ecopover  8763  frins  9671  alephsuc2  9997  indstr  12861  reltxrnmnf  13290  ackbijnn  15788  mrelatglb0  18522  0frgp  19749  iccpnfcnv  24925  prter2  39345  natlocalincr  47326  natglobalincr  47327