Theorem ineq12 4178

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 4176	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 4177	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2784	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3913

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-in 3921

This theorem is referenced by:  ineq12i  4181  ineq12d  4184  ineqan12d  4185  fnun  6632  frrlem4  8268  undifixp  8907  endisj  9028  sbthlem8  9058  fiin  9373  pm54.43  9954  kmlem9  10112  indistopon  22888  epttop  22896  restbas  23045  ordtbas2  23078  txbas  23454  ptbasin  23464  trfbas2  23730  snfil  23751  fbasrn  23771  trfil2  23774  fmfnfmlem3  23843  ustuqtop2  24130  minveclem3b  25328  isperp  28639  brredunds  38617  diophin  42760  kelac2lem  43053  iscnrm3r  48936  incat  49590  setc1onsubc  49591