Theorem ineq12 4169

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 4167	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 4168	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2792	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∩ cin 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-in 3910

This theorem is referenced by:  ineq12i  4172  ineq12d  4175  ineqan12d  4176  fnun  6614  frrlem4  8241  undifixp  8884  endisj  9004  sbthlem8  9034  fiin  9337  pm54.43  9925  kmlem9  10081  indistopon  22957  epttop  22965  restbas  23114  ordtbas2  23147  txbas  23523  ptbasin  23533  trfbas2  23799  snfil  23820  fbasrn  23840  trfil2  23843  fmfnfmlem3  23912  ustuqtop2  24198  minveclem3b  25396  isperp  28796  brredunds  38961  eldisjim3  39066  diophin  43129  kelac2lem  43421  iscnrm3r  49307  incat  49960  setc1onsubc  49961