Theorem eqvf 3432

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 2906	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	cleqf 2937	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
5	4	tbt 369	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
6	5	albii 1823	. 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 2108  Ⅎwnfc 2886  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424

This theorem is referenced by: (None)