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  6308  wfis2f  6310  wfis2  6312  isarep2  6580  mpoexw  8022  ecopover  8759  frins  9665  alephsuc2  9991  indstr  12855  reltxrnmnf  13284  ackbijnn  15782  mrelatglb0  18516  0frgp  19743  iccpnfcnv  24920  prter2  39338  natlocalincr  47319  natglobalincr  47320