Theorem difeq2d 1588

Description: Deduction adding difference to the left in a class equality.

Hypothesis

Ref	Expression
difeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
difeq2d	⊢ (φ → (C ∖ A) = (C ∖ B))

Proof of Theorem difeq2d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 ⊢ (φ → A = B)
2		difeq2 1583	. 2 ⊢ (A = B → (C ∖ A) = (C ∖ B))
3	1, 2	syl 12	1 ⊢ (φ → (C ∖ A) = (C ∖ B))

Syntax hints:   → wi 2   = wceq 1091   ∖ cdif 1484

This theorem is referenced by:  tz7.49 2997  sbthlem2 3350  sbthlem3 3351  sbth 3359  phplem4 3406  unblem2 3432  unblem3 3433  kmlem8 3587  kmlem10 3589  kmlem11 3590  alephsuc3 4955

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-dif 1489

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