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Theorem ralcomf 2592
 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 1416 . . . 4 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ((𝑦𝐵𝑥𝐴) → 𝜑))
212albii 1447 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑))
3 alcom 1454 . . 3 (∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
42, 3bitri 183 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
5 ralcomf.1 . . 3 𝑦𝐴
65r2alf 2452 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
7 ralcomf.2 . . 3 𝑥𝐵
87r2alf 2452 . 2 (∀𝑦𝐵𝑥𝐴 𝜑 ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
94, 6, 83bitr4i 211 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   ∈ wcel 1480  Ⅎwnfc 2268  ∀wral 2416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421 This theorem is referenced by:  ralcom  2594  ssiinf  3862
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