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Theorem ralcom4 2593
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 2490 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥𝐴 𝜑)
2 ralv 2588 . . 3 (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
32ralbii 2347 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑥𝐴𝑦𝜑)
4 ralv 2588 . 2 (∀𝑦 ∈ V ∀𝑥𝐴 𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
51, 3, 43bitr3i 203 1 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257  wral 2323  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by:  uniiunlem  3056  uni0b  3633  iunss  3726  trint  3897  reliun  4486  funimass4  5252  ralrnmpt2  5643
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