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Theorem ralv 2586
Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
ralv (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem ralv
StepHypRef Expression
1 df-ral 2326 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 2575 . . . 4 𝑥 ∈ V
32a1bi 236 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1373 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 180 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1255  wcel 1407  wral 2321  Vcvv 2572
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-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-ral 2326  df-v 2574
This theorem is referenced by:  ralcom4  2591  viin  3741  issref  4732
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