Theorem ineq2i 3375

Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Hypothesis

Ref	Expression
ineq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
ineq2i	⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)

Proof of Theorem ineq2i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		ineq2 3372	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)

Syntax hints:   = wceq 1373   ∩ cin 3169

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176

This theorem is referenced by:  in4  3393  inindir  3395  indif2  3421  difun1  3437  dfrab3ss  3455  dfif3  3589  intunsn  3932  rint0  3933  riin0  4008  res0  4977  resres  4985  resundi  4986  resindi  4988  inres  4990  resiun2  4993  resopab  5017  dfse2  5069  dminxp  5141  imainrect  5142  resdmres  5188  funimacnv  5364  unfiin  7044  sbthlemi5  7084  dmaddpi  7468  dmmulpi  7469  fsumiun  11873  ressval2  12983  ressval3d  12989  lgsquadlem3  15641