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  4239  unidif0  4251  unipw  4303  snnex  4539  unon  4603  xpiundi  4777  xpiundir  4778  0xp  4799  iunxpf  4870  cnvuni  4908  dmiun  4932  dmuni  4933  epini  5099  rniun  5139  cnvresima  5218  imaco  5234  rnco  5235  dfmpt3  5446  imaiun  5890  opabex3d  6272  opabex3  6273  ecid  6753  qsid  6755  mapval2  6833  ixpin  6878  dfz2  9527  infssuzex  10461  dfrp2  10491  1nprm  12644  infpn2  13035  rrgval  14234  2idlval  14474  cnfldui  14561  zrhval  14589  plyun0  15418