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Theorem ralcom4 2677
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
ralcom4 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem ralcom4
StepHypRef Expression
1 ralcom 2566 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥𝐴 𝜑)
2 ralv 2672 . . 3 (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
32ralbii 2413 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑥𝐴𝑦𝜑)
4 ralv 2672 . 2 (∀𝑦 ∈ V ∀𝑥𝐴 𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 209 1 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1310  wral 2388  Vcvv 2655
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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657
This theorem is referenced by:  uniiunlem  3149  uni0b  3725  iunss  3818  disjnim  3884  trint  3999  reliun  4618  funimass4  5424  ralrnmpo  5837
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