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Theorem aaanh 1548
 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 1524 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1429 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaanh.2 . . . 4 (𝜓 → ∀𝑥𝜓)
54hbal 1436 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
6519.27h 1522 . 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 1312 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-4 1470 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  mo23  2016  2eu4  2068
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