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Theorem aaan 2331
Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2140. (Revised by Gino Giotto, 21-Nov-2024.)
Hypotheses
Ref Expression
aaan.1 𝑦𝜑
aaan.2 𝑥𝜓
Assertion
Ref Expression
aaan (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Proof of Theorem aaan
StepHypRef Expression
1 19.26-2 1877 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))
2 aaan.1 . . . . 5 𝑦𝜑
3219.3 2198 . . . 4 (∀𝑦𝜑𝜑)
43albii 1825 . . 3 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝜑)
5 alcom 2159 . . . 4 (∀𝑥𝑦𝜓 ↔ ∀𝑦𝑥𝜓)
6 aaan.2 . . . . . 6 𝑥𝜓
7619.3 2198 . . . . 5 (∀𝑥𝜓𝜓)
87albii 1825 . . . 4 (∀𝑦𝑥𝜓 ↔ ∀𝑦𝜓)
95, 8bitri 274 . . 3 (∀𝑥𝑦𝜓 ↔ ∀𝑦𝜓)
104, 9anbi12i 626 . 2 ((∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
111, 10bitri 274 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wal 1539  wnf 1789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-11 2157  ax-12 2174
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790
This theorem is referenced by:  aaanv  41959  pm11.71  41968
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