Theorem difeq1d 3132

Description: Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Hypothesis

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

Assertion

Ref	Expression
difeq1d	⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Proof of Theorem difeq1d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		difeq1 3126	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Syntax hints:   → wi 4   = wceq 1296   ∖ cdif 3010

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-dif 3015

This theorem is referenced by:  difeq12d  3134  diftpsn3  3600  phplem4  6651  phplem3g  6652  phplem4on  6663  en2other2  6919  isstruct2im  11669  isstruct2r  11670  setsfun0  11695  cldval  11967  difopn  11976  cnclima  12090