Theorem ineq2i 1642

Description: Equality inference for intersection of two classes.

Hypothesis

Ref	Expression
ineq1i.1	⊢ A = B

Assertion

Ref	Expression
ineq2i	⊢ (C ∩ A) = (C ∩ B)

Proof of Theorem ineq2i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ A = B
2		ineq2 1639	. 2 ⊢ (A = B → (C ∩ A) = (C ∩ B))
3	1, 2	ax-mp 6	1 ⊢ (C ∩ A) = (C ∩ B)

Syntax hints:   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  ineq12i 1643  in23 1652  in4 1653  inindir 1655  difun1 1687  difab 1693  difin0 1759  undif1 1761  difdifdir 1765  intunsn 1993  dfepfr 2184  epfrc 2185  res0 2578  resundi 2582  resopab 2598  dffr3 2620  funimacnv 2711  tz7.44-2 2967  tz7.44-3 2968  frfnom 2989  sbthlem5 3353  zfregs 3491  kmlem10 3589  dmaddpi 3812  dmmulpi 3813  chdmj3 5404  chdmj4 5405  chjass 5407  pjoml2 5495  pjoml3 5496  cmcmlem 5500  cmcm2 5502  cmbr3 5509  fh1 5518  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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