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Theorem eqvf 3177
 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 2751 . . 3 𝑥V
31, 2cleqf 2776 . 2 (𝐴 = V ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
4 vex 3176 . . . 4 𝑥 ∈ V
54tbt 358 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ V))
65albii 1737 . 2 (∀𝑥 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ V))
73, 6bitr4i 266 1 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175 This theorem is referenced by:  eqv  3178
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