Theorem ineq12i 1643

Description: Equality inference for intersection of two classes.

Hypotheses

Ref	Expression
ineq1i.1	⊢ A = B
ineq12i.2	⊢ C = D

Assertion

Ref	Expression
ineq12i	⊢ (A ∩ C) = (B ∩ D)

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. . 3 ⊢ A = B
2	1	ineq1i 1641	. 2 ⊢ (A ∩ C) = (B ∩ C)
3		ineq12i.2	. . 3 ⊢ C = D
4	3	ineq2i 1642	. 2 ⊢ (B ∩ C) = (B ∩ D)
5	2, 4	eqtr 1119	1 ⊢ (A ∩ C) = (B ∩ D)

Syntax hints:   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  difundi 1681  difindir 1684  rnin 2645  fodomb 3615  cmbr3 5509  fh1 5518  3oalem5 5556

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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