Theorem eqv 3428

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 2159	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
2		vex 2729	. . . 4 ⊢ 𝑥 ∈ V
3	2	tbt 246	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4	3	albii 1458	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
5	1, 4	bitr4i 186	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  setindel  4515  dmi  4819