Theorem eqvf 3442

Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
eqvf.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
eqvf	⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Proof of Theorem eqvf

Step	Hyp	Ref	Expression
1		eqvf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2907	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	cleqf 2938	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4		vex 3436	. . . 4 ⊢ 𝑥 ∈ V
5	4	tbt 370	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
6	5	albii 1822	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
7	3, 6	bitr4i 277	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434

This theorem is referenced by: (None)