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Theorem 19.21-2 1557
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 1475 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
32albii 1359 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4 19.21-2.1 . . 3 𝑥𝜑
5419.21 1475 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
63, 5bitri 173 1 (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241  wnf 1349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by: (None)
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