Theorem eqvisset 3498

Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3492 and issetri 3497. (Contributed by BJ, 27-Apr-2019.)

Assertion

Ref	Expression
eqvisset	⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

Proof of Theorem eqvisset

Step	Hyp	Ref	Expression
1		vex 3482	. 2 ⊢ 𝑥 ∈ V
2		eleq1 2827	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
3	1, 2	mpbii 233	1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480

This theorem is referenced by:  ceqex  3652  moeq3  3721  mo2icl  3723  eusvnfb  5399  oprabv  7493  elxp5  7946  xpsnen  9094  fival  9450  dffi2  9461  tz9.12lem1  9825  m1detdiag  22619  dvfsumlem1  26081  dchrisumlema  27547  dchrisumlem2  27549  fnimage  35911  bj-csbsnlem  36886  copsex2b  37123  pr2cv  43538  disjf1o  45134  mptssid  45185  fourierdlem49  46111