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Theorem ralcom4f 29625
 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 2902 . . 3 𝑥V
31, 2ralcomf 3234 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥𝐴 𝜑)
4 ralv 3359 . . 3 (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
54ralbii 3118 . 2 (∀𝑥𝐴𝑦 ∈ V 𝜑 ↔ ∀𝑥𝐴𝑦𝜑)
6 ralv 3359 . 2 (∀𝑦 ∈ V ∀𝑥𝐴 𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
73, 5, 63bitr3i 290 1 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wal 1630  Ⅎwnfc 2889  ∀wral 3050  Vcvv 3340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342 This theorem is referenced by: (None)
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