Theorem ineq12 4141

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 4139	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		ineq2 4140	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐷))
3	1, 2	sylan9eq 2798	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-in 3894

This theorem is referenced by:  ineq12i  4144  ineq12d  4147  ineqan12d  4148  fnun  6545  frrlem4  8105  undifixp  8722  endisj  8845  sbthlem8  8877  fiin  9181  pm54.43  9759  kmlem9  9914  indistopon  22151  epttop  22159  restbas  22309  ordtbas2  22342  txbas  22718  ptbasin  22728  trfbas2  22994  snfil  23015  fbasrn  23035  trfil2  23038  fmfnfmlem3  23107  ustuqtop2  23394  minveclem3b  24592  isperp  27073  brredunds  36739  diophin  40594  kelac2lem  40889  iscnrm3r  46242