Theorem ineq2d 3432

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
ineq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ineq2d	⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Proof of Theorem ineq2d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ineq2 3426	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1402   ∩ cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  disjpr2  3769  rint0  4004  riin0  4079  disji2  4117  xpriindim  4913  riinint  5038  reseq2  5053  csbresg  5061  resindm  5100  isoselem  6016  zfz1isolem1  11270  fsumm1  12161  bitsinv1  12707  ballotfilemfval  13207  ennnfonelemhf1o  13282  nninfdclemcl  13317  nninfdclemp1  13319  nninfdc  13322  ressvalsets  13395  ressbasd  13398  ressinbasd  13405  ressressg  13406  restval  13576  mgpress  14205  subrngpropd  14497  subrgpropd  14534  crng2idl  14840  basis1  15071  baspartn  15074  eltg  15076  tgdom  15096  ntrval  15134  resttopon2  15202  restopnb  15205  qtopbasss  15545  p1evtxdeqfilem  16466