Theorem ineq1i 1641

Description: Equality inference for intersection of two classes.

Hypothesis

Ref	Expression
ineq1i.1	⊢ A = B

Assertion

Ref	Expression
ineq1i	⊢ (A ∩ C) = (B ∩ C)

Proof of Theorem ineq1i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ A = B
2		ineq1 1638	. 2 ⊢ (A = B → (A ∩ C) = (B ∩ C))
3	1, 2	ax-mp 6	1 ⊢ (A ∩ C) = (B ∩ C)

Syntax hints:   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  ineq12i 1643  in12 1651  inindi 1654  dfrab2 1696  rescom 2588  resabs1 2592  resabs2 2593  resdisj 2656  ssenen 3399  chdmj2 5403  chjass 5407  pjoml2 5495  pjoml4 5497  cmcmlem 5500  fh2 5519

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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