Theorem inteqi 3640

Description: Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)

Hypothesis

Ref	Expression
inteqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
inteqi	⊢ ∩ 𝐴 = ∩ 𝐵

Proof of Theorem inteqi

Step	Hyp	Ref	Expression
1		inteqi.1	. 2 ⊢ 𝐴 = 𝐵
2		inteq 3639	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)
3	1, 2	ax-mp 7	1 ⊢ ∩ 𝐴 = ∩ 𝐵

Syntax hints:   = wceq 1284  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-int 3637

This theorem is referenced by:  elintrab  3648  ssintrab  3659  intmin2  3662  intsng  3670  intexrabim  3928  op1stb  4227  bm2.5ii  4240  dfiin3g  4608  op2ndb  4824  bj-dfom  10728  bj-omind  10729