Theorem ineq12 4181

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 4179	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 4180	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2785	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-in 3924

This theorem is referenced by:  ineq12i  4184  ineq12d  4187  ineqan12d  4188  fnun  6635  frrlem4  8271  undifixp  8910  endisj  9032  sbthlem8  9064  fiin  9380  pm54.43  9961  kmlem9  10119  indistopon  22895  epttop  22903  restbas  23052  ordtbas2  23085  txbas  23461  ptbasin  23471  trfbas2  23737  snfil  23758  fbasrn  23778  trfil2  23781  fmfnfmlem3  23850  ustuqtop2  24137  minveclem3b  25335  isperp  28646  brredunds  38624  diophin  42767  kelac2lem  43060  iscnrm3r  48940  incat  49594  setc1onsubc  49595