Theorem difeq2d 3255

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 3249	. 2 |- ( A = B -> ( C \ A ) = ( C \ B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -> wi 4   = wceq 1353   \ cdif 3128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rab 2464  df-dif 3133

This theorem is referenced by:  difeq12d  3256  exmid1stab  4210  phplem3  6856  phplem4  6857  phplem3g  6858  phplem4dom  6864  phplem4on  6869  fidifsnen  6872  xpfi  6931  sbthlem2  6959  sbthlemi3  6960  isbth  6968  ismkvnex  7155  setsvalg  12494  setsvala  12495