Theorem eqvisset 3184

Description: A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 3180 and issetri 3183. (Contributed by BJ, 27-Apr-2019.)

Assertion

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

Proof of Theorem eqvisset

Step	Hyp	Ref	Expression
1		vex 3176	. 2 ⊢ 𝑥 ∈ V
2		eleq1 2676	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
3	1, 2	mpbii 222	1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  ceqex  3303  moeq3  3350  mo2icl  3352  eusvnfb  4788  oprabv  6601  elxp5  7004  xpsnen  7929  fival  8201  dffi2  8212  tz9.12lem1  8533  m1detdiag  20222  dvfsumlem1  23593  dchrisumlema  24977  dchrisumlem2  24979  fnimage  31206  bj-csbsnlem  32090  disjf1o  38373  fourierdlem49  39048