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Theorem ralcomf 2542
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 1381 . . . 4 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ((𝑦𝐵𝑥𝐴) → 𝜑))
212albii 1412 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑))
3 alcom 1419 . . 3 (∀𝑥𝑦((𝑦𝐵𝑥𝐴) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
42, 3bitri 183 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦𝑥((𝑦𝐵𝑥𝐴) → 𝜑))
5 ralcomf.1 . . 3 𝑦𝐴
65r2alf 2406 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
7 ralcomf.2 . . 3 𝑥𝐵
87r2alf 2406 . 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 1294  wcel 1445  wnfc 2222  wral 2370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375
This theorem is referenced by:  ralcom  2544  ssiinf  3801
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