Theorem ineq2 1639

Description: Equality theorem for intersection of two classes.

Assertion

Ref	Expression
ineq2	⊢ (A = B → (C ∩ A) = (C ∩ B))

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 1638	. 2 ⊢ (A = B → (A ∩ C) = (B ∩ C))
2		incom 1636	. 2 ⊢ (C ∩ A) = (A ∩ C)
3		incom 1636	. 2 ⊢ (C ∩ B) = (B ∩ C)
4	1, 2, 3	3eqtr4g 1147	1 ⊢ (A = B → (C ∩ A) = (C ∩ B))

Syntax hints:   → wi 2   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  ineq12 1640  ineq2i 1642  ineq2d 1645  wefrc 2195  onfr 2237  fiint 3445  cplem2 3546  aceq5 3563  kmlem2 3581  kmlem12 3591  kmlem14 3593  shinclt 5352  chinclt 5416  chdmm1t 5438  cmbrt 5494  cmbr3 5509  stcltrlem1 5709  mdbr 5726  cvexcht 5763  sumdmdi 5785

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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