Theorem eqvisset 3456

Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3450 and issetri 3455. (Contributed by BJ, 27-Apr-2019.)

Assertion

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

Proof of Theorem eqvisset

Step	Hyp	Ref	Expression
1		vex 3440	. 2 ⊢ 𝑥 ∈ V
2		eleq1 2819	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
3	1, 2	mpbii 233	1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438

This theorem is referenced by:  ceqex  3602  moeq3  3666  mo2icl  3668  eusvnfb  5326  oprabv  7401  elxp5  7848  xpsnen  8969  fival  9291  dffi2  9302  tz9.12lem1  9675  m1detdiag  22507  dvfsumlem1  25954  dchrisumlema  27421  dchrisumlem2  27423  fnimage  35963  bj-csbsnlem  36937  copsex2b  37174  pr2cv  43581  disjf1o  45228  mptssid  45278  fourierdlem49  46193