Theorem ineq2i 3419

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 3416	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)

Syntax hints:   = wceq 1398   ∩ cin 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217

This theorem is referenced by:  in4  3437  inindir  3439  indif2  3465  difun1  3481  dfrab3ss  3499  dfif3  3636  intunsn  3987  rint0  3988  riin0  4063  res0  5042  resres  5050  resundi  5051  resindi  5053  inres  5055  resiun2  5058  resopab  5082  dfse2  5135  dminxp  5207  imainrect  5208  resdmres  5254  funimacnv  5432  unfiin  7186  sbthlemi5  7231  dmaddpi  7640  dmmulpi  7641  hashtpgim  11217  fsumiun  12163  ressval2  13279  ressval3d  13285  lgsquadlem3  15952