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Theorem ralcomf 2651
Description: Commutation of restricted quantifiers. (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 ancomsimp 1451 . . . 4 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ((𝑦𝐵𝑥𝐴) → 𝜑))
212albii 1482 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑))
3 alcom 1489 . . 3 (∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
42, 3bitri 184 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
5 ralcomf.1 . . 3 𝑦𝐴
65r2alf 2507 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
7 ralcomf.2 . . 3 𝑥𝐵
87r2alf 2507 . 2 (∀𝑦𝐵𝑥𝐴 𝜑 ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
94, 6, 83bitr4i 212 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362  wcel 2160  wnfc 2319  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  ralcom  2653  ssiinf  3951
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