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Theorem rspec2 3175
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 3172 . 2 (𝑥𝐴 → ∀𝑦𝐵 𝜑)
32r19.21bi 3173 1 ((𝑥𝐴𝑦𝐵) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  wral 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111
This theorem is referenced by:  rspec3  3176
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