Theorem rspec 3242

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 3239	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2099  ∀wral 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2164

This theorem depends on definitions:  df-bi 206  df-ex 1775  df-ral 3057

This theorem is referenced by:  rspec2  3271  vtoclri  3571  wfis  6355  wfis2f  6358  wfis2  6360  isarep2  6638  mpoexw  8077  ecopover  8831  frins  9767  alephsuc2  10095  indstr  12922  reltxrnmnf  13345  ackbijnn  15798  mrelatglb0  18544  0frgp  19725  iccpnfcnv  24856  prter2  38290  natlocalincr  46185  natglobalincr  46186