Theorem eqvf 3473

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 2892	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	cleqf 2924	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4		vex 3466	. . . 4 ⊢ 𝑥 ∈ V
5	4	tbt 368	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
6	5	albii 1814	. 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 1532   = wceq 1534   ∈ wcel 2099  Ⅎwnfc 2876  Vcvv 3462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-v 3464

This theorem is referenced by: (None)