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Theorem aaanh 1519
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
Hypotheses
Ref Expression
aaanh.1 (𝜑 → ∀𝑦𝜑)
aaanh.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
aaanh (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Proof of Theorem aaanh
StepHypRef Expression
1 aaanh.1 . . . 4 (𝜑 → ∀𝑦𝜑)
2119.28h 1495 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1400 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaanh.2 . . . 4 (𝜓 → ∀𝑥𝜓)
54hbal 1407 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
6519.27h 1493 . 2 (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
73, 6bitri 182 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  mo23  1984  2eu4  2036
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