Theorem difeq1d 3346

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 3340	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∖ cdif 3217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-dif 3222

This theorem is used by:  difeq12d  3348  diftpsn3  3856  phplem4  7162  phplem3g  7163  phplem4on  7175  isfsuppfi  7296  funisfsuppdc  7298  snopfsupp  7306  en2other2  7554  ballotfilemfval  13286  ballotfilemfp1  13288  ballotfilemfc0  13289  ballotfilemfcc  13290  ballotfilemgval  13324  ballotfilemgun  13325  isstruct2im  13419  isstruct2r  13420  setsfun0  13445  ptex  13676  mvrclfi  15226  mplsubgfilemcl  15232  mplsubgfileminv  15233  cldval  15342  difopn  15351  cnclima  15466