MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspec2 Structured version   Visualization version   GIF version

Theorem rspec2 3208
Description: Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.)
Hypothesis
Ref Expression
rspec2.1 𝑥𝐴𝑦𝐵 𝜑
Assertion
Ref Expression
rspec2 ((𝑥𝐴𝑦𝐵) → 𝜑)

Proof of Theorem rspec2
StepHypRef Expression
1 rspec2.1 . . 3 𝑥𝐴𝑦𝐵 𝜑
21rspec 3204 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32r19.21bi 3205 1 ((𝑥𝐴𝑦𝐵) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-ral 3140
This theorem is referenced by:  rspec3  3209
  Copyright terms: Public domain W3C validator