Theorem ineq2d 1645

Description: Equality deduction for intersection of two classes.

Hypothesis

Ref	Expression
ineq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
ineq2d	⊢ (φ → (C ∩ A) = (C ∩ B))

Proof of Theorem ineq2d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (φ → A = B)
2		ineq2 1639	. 2 ⊢ (A = B → (C ∩ A) = (C ∩ B))
3	1, 2	syl 12	1 ⊢ (φ → (C ∩ A) = (C ∩ B))

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

This theorem is referenced by:  ineq12d 1646  frirr 2176  fr2nr 2177  fr3nr 2178  reseq2 2576  resabs1 2592  resabs2 2593  resdisj 2656  isofrlem 2939  kmlem10 3589  omls 5251  chdmj3t 5444  chdmj4t 5445  cmbrt 5494  pjoml3t 5517  dmdbr 5731

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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