Theorem ineq12 4162

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Assertion

Ref	Expression
ineq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 4160	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 4161	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2786	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∩ cin 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-in 3904

This theorem is referenced by:  ineq12i  4165  ineq12d  4168  ineqan12d  4169  fnun  6595  frrlem4  8219  undifixp  8858  endisj  8977  sbthlem8  9007  fiin  9306  pm54.43  9894  kmlem9  10050  indistopon  22916  epttop  22924  restbas  23073  ordtbas2  23106  txbas  23482  ptbasin  23492  trfbas2  23758  snfil  23779  fbasrn  23799  trfil2  23802  fmfnfmlem3  23871  ustuqtop2  24157  minveclem3b  25355  isperp  28690  brredunds  38732  diophin  42875  kelac2lem  43167  iscnrm3r  49058  incat  49712  setc1onsubc  49713