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Theorem ralcomf 3293
Description: Commutation of restricted universal quantifiers. For a version based on fewer axioms see ralcom 3280. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
ralcomf.1 𝑦𝐴
ralcomf.2 𝑥𝐵
Assertion
Ref Expression
ralcomf (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem ralcomf
StepHypRef Expression
1 ancomst 464 . . . 4 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ((𝑦𝐵𝑥𝐴) → 𝜑))
212albii 1814 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑))
3 alcom 2148 . . 3 (∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
42, 3bitri 275 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
5 ralcomf.1 . . 3 𝑦𝐴
65r2alf 3272 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
7 ralcomf.2 . . 3 𝑥𝐵
87r2alf 3272 . 2 (∀𝑦𝐵𝑥𝐴 𝜑 ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
94, 6, 83bitr4i 303 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531  wcel 2098  wnfc 2877  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-11 2146  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-clel 2804  df-nfc 2879  df-ral 3056
This theorem is referenced by:  ssiinf  5050  ralcom4f  32213
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