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Theorem eqvf 3480
 Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
eqvf.1 𝑥𝐴
Assertion
Ref Expression
eqvf (𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Proof of Theorem eqvf
StepHypRef Expression
1 eqvf.1 . . 3 𝑥𝐴
2 nfcv 2974 . . 3 𝑥V
31, 2cleqf 3003 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
4 vex 3474 . . . 4 𝑥 ∈ V
54tbt 373 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
65albii 1821 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
73, 6bitr4i 281 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2115  Ⅎwnfc 2958  Vcvv 3471 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 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473 This theorem is referenced by: (None)
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