Theorem difeq2d 3265

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 3259	. 2 |- ( A = B -> ( C \ A ) = ( C \ B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -> wi 4   = wceq 1363   \ cdif 3138

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 615  ax-in2 616  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-rab 2474  df-dif 3143

This theorem is referenced by:  difeq12d  3266  exmid1stab  4220  phplem3  6868  phplem4  6869  phplem3g  6870  phplem4dom  6876  phplem4on  6881  fidifsnen  6884  xpfi  6943  sbthlem2  6971  sbthlemi3  6972  isbth  6980  ismkvnex  7167  setsvalg  12506  setsvala  12507