Theorem rspec 3255

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 3252	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1813  df-ral 3079

This theorem is used by:  rspec2  3283  vtoclri  3547  rab0  4338  wfis  6354  wfis2f  6356  wfis2  6358  isarep2  6626  mpoexw  8080  ecopover  8824  frins  9737  alephsuc2  10086  indstr  12966  reltxrnmnf  13395  ackbijnn  15917  mrelatglb0  18651  0frgp  19905  iccpnfcnv  25171  prter2  39739