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Theorem 19.21-2 1645
 Description: Theorem 19.21 of [Margaris] p. 90 but with 2 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 1562 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
32albii 1446 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4 19.21-2.1 . . 3 𝑥𝜑
5419.21 1562 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
63, 5bitri 183 1 (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437 This theorem is referenced by: (None)
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