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Theorem r2alf 3281
Description: Double restricted universal quantification. For a version based on fewer axioms see r2al 3195. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3142. (Revised by Wolf Lammen, 9-Jan-2020.)
Hypothesis
Ref Expression
r2alf.1 𝑦𝐴
Assertion
Ref Expression
r2alf (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2alf
StepHypRef Expression
1 r2alf.1 . . . 4 𝑦𝐴
21nfcri 2897 . . 3 𝑦 𝑥𝐴
3219.21 2207 . 2 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
43r2allem 3142 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  wnfc 2890  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-clel 2816  df-nfc 2892  df-ral 3062
This theorem is referenced by:  r2exf  3282  ralcomf  3302
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