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Theorem ralv 2571
 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 2311 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 2560 . . . 4 𝑥 ∈ V
32a1bi 232 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1359 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 176 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  ∀wral 2306  Vcvv 2557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-v 2559 This theorem is referenced by:  ralcom4  2576  viin  3716  issref  4707
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