Theorem eqvisset 3474

Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3468 and issetri 3473. (Contributed by BJ, 27-Apr-2019.)

Assertion

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

Proof of Theorem eqvisset

Step	Hyp	Ref	Expression
1		vex 3458	. 2 ⊢ 𝑥 ∈ V
2		eleq1 2852	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
3	1, 2	mpbii 236	1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  ceqex  3609  moeq3  3673  mo2icl  3675  eusvnfb  5362  oprabv  7511  elxp5  7969  xpsnen  9118  fival  9441  dffi2  9452  tz9.12lem1  9831  m1detdiag  22967  dvfsumlem1  26401  dchrisumlema  27872  dchrisumlem2  27874  oldfib  28820  fnimage  36906  bj-csbsnlem  38030  copsex2b  38332  pr2cv  44810  disjf1o  46479  mptssid  46526  fourierdlem49  47438