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Theorem 19.21-2 2065
 Description: Version of 19.21 2062 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
Hypotheses
Ref Expression
19.21-2.1 𝑥𝜑
19.21-2.2 𝑦𝜑
Assertion
Ref Expression
19.21-2 (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Proof of Theorem 19.21-2
StepHypRef Expression
1 19.21-2.2 . . . 4 𝑦𝜑
2119.21 2062 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
32albii 1737 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4 19.21-2.1 . . 3 𝑥𝜑
5419.21 2062 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
63, 5bitri 263 1 (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  cotr2g  13563  dford4  36614
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