Theorem eqvisset 3460

Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3454 and issetri 3459. (Contributed by BJ, 27-Apr-2019.)

Assertion

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

Proof of Theorem eqvisset

Step	Hyp	Ref	Expression
1		vex 3444	. 2 ⊢ 𝑥 ∈ V
2		eleq1 2824	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
3	1, 2	mpbii 233	1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440

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 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442

This theorem is referenced by:  ceqex  3606  moeq3  3670  mo2icl  3672  eusvnfb  5338  oprabv  7418  elxp5  7865  xpsnen  8989  fival  9315  dffi2  9326  tz9.12lem1  9699  m1detdiag  22541  dvfsumlem1  25988  dchrisumlema  27455  dchrisumlem2  27457  oldfib  28373  fnimage  36121  bj-csbsnlem  37104  copsex2b  37345  pr2cv  43789  disjf1o  45435  mptssid  45485  fourierdlem49  46399