Theorem ineq12 3771

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 3769	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 3770	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2664	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547

This theorem is referenced by:  ineq12i  3774  ineq12d  3777  ineqan12d  3778  fnun  5911  undifixp  7830  endisj  7932  sbthlem8  7962  fiin  8211  pm54.43  8709  kmlem9  8863  indistopon  20615  epttop  20623  restbas  20772  ordtbas2  20805  txbas  21180  ptbasin  21190  trfbas2  21457  snfil  21478  fbasrn  21498  trfil2  21501  fmfnfmlem3  21570  ustuqtop2  21856  minveclem3b  23007  isperp  25407  frrlem4  31027  diophin  36354  kelac2lem  36652