Theorem difeq2d 3347

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

Hypothesis

Ref	Expression
difeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
difeq2d	|- ( ph -> ( C \ A ) = ( C \ B ) )

Proof of Theorem difeq2d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 |- ( ph -> A = B )
2		difeq2 3341	. 2 |- ( A = B -> ( C \ A ) = ( C \ B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C \ A ) = ( C \ B ) )

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-in1 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  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-ral 2533  df-rab 2537  df-dif 3222

This theorem is used by:  difeq12d  3348  exmid1stab  4347  phplem3  7159  phplem4  7160  phplem3g  7161  phplem4dom  7167  phplem4on  7173  fidifsnen  7176  xpfi  7243  sbthlem2  7280  sbthlemi3  7281  isbth  7289  ismkvnex  7501  ballotfilemfval  13287  setsvalg  13440  setsvala  13441  wexmiddiffilem  17248  wexmiddifxylem  17250