Theorem eqriv 2226

Description: Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
eqriv.1	⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
eqriv	⊢ 𝐴 = 𝐵

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqriv

Step	Hyp	Ref	Expression
1		dfcleq 2223	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2		eqriv.1	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	1, 2	mpgbir 1499	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1495  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eqid  2229  sb8ab  2351  cbvabw  2352  cbvab  2353  vjust  2800  nfccdeq  3026  csbcow  3135  difeqri  3324  uneqri  3346  incom  3396  ineqri  3397  difin  3441  invdif  3446  indif  3447  difundi  3456  indifdir  3460  pwv  3887  uniun  3907  intun  3954  intpr  3955  iuncom  3971  iuncom4  3972  iun0  4022  0iun  4023  iunin2  4029  iunun  4044  iunxun  4045  iunxiun  4047  iinpw  4056  inuni  4240  unidif0  4252  unipw  4304  snnex  4540  unon  4604  xpiundi  4779  xpiundir  4780  0xp  4801  iunxpf  4873  cnvuni  4911  dmiun  4935  dmuni  4936  epini  5102  rniun  5142  cnvresima  5221  imaco  5237  rnco  5238  dfmpt3  5449  imaiun  5893  opabex3d  6275  opabex3  6276  ecid  6758  qsid  6760  mapval2  6838  ixpin  6883  dfz2  9535  infssuzex  10470  dfrp2  10500  1nprm  12657  infpn2  13048  rrgval  14247  2idlval  14487  cnfldui  14574  zrhval  14602  plyun0  15431  clwwlkn0  16177