Theorem rspec 3248

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 3245	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-ral 3063

This theorem is referenced by:  rspec2  3277  vtoclri  3577  wfis  6357  wfis2f  6360  wfis2  6362  isarep2  6640  mpoexw  8065  ecopover  8815  frins  9747  alephsuc2  10075  indstr  12900  reltxrnmnf  13321  ackbijnn  15774  mrelatglb0  18514  0frgp  19647  iccpnfcnv  24460  prter2  37751  natlocalincr  45590  natglobalincr  45591