Theorem rspec 3226

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 3223	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3044

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  df-ral 3045

This theorem is referenced by:  rspec2  3254  vtoclri  3553  wfis  6313  wfis2f  6315  wfis2  6317  isarep2  6590  mpoexw  8036  ecopover  8771  frins  9681  alephsuc2  10009  indstr  12851  reltxrnmnf  13279  ackbijnn  15770  mrelatglb0  18502  0frgp  19693  iccpnfcnv  24875  prter2  38867  natlocalincr  46867  natglobalincr  46868