Theorem ineq12 1640

Description: Equality theorem for intersection of two classes.

Assertion

Ref	Expression
ineq12	⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 1638	. 2 ⊢ (A = B → (A ∩ C) = (B ∩ C))
2		ineq2 1639	. 2 ⊢ (C = D → (B ∩ C) = (B ∩ D))
3	1, 2	sylan9eq 1144	1 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  ssin 1659  undir 1679  fnun 2730  endisj 3341  sbthlem8 3356  kmlem8 3587  infxpidmlem11 4943  5oa 5551

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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