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Theorem aaanh 1573
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 1549 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1457 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaanh.2 . . . 4 (𝜓 → ∀𝑥𝜓)
54hbal 1464 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
6519.27h 1547 . 2 (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
73, 6bitri 183 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1340
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 1434  ax-7 1435  ax-gen 1436  ax-4 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  mo23  2054  2eu4  2106
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