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Theorem ralcom4f 30249
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
Hypothesis
Ref Expression
ralcom4f.1 𝑦𝐴
Assertion
Ref Expression
ralcom4f (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem ralcom4f
StepHypRef Expression
1 ralcom4f.1 . . 3 𝑦𝐴
2 nfcv 2982 . . 3 𝑥V
31, 2ralcomf 3348 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥𝐴 𝜑)
4 ralv 3505 . . 3 (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
54ralbii 3160 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑥𝐴𝑦𝜑)
6 ralv 3505 . 2 (∀𝑦 ∈ V ∀𝑥𝐴 𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
73, 5, 63bitr3i 304 1 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wnfc 2962  wral 3133  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482
This theorem is referenced by: (None)
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