Theorem difeq12d 3326

Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Hypotheses

Ref	Expression
difeq12d.1	|- ( ph -> A = B )
difeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
difeq12d	|- ( ph -> ( A \ C ) = ( B \ D ) )

Proof of Theorem difeq12d

Step	Hyp	Ref	Expression
1		difeq12d.1	. . 3 |- ( ph -> A = B )
2	1	difeq1d 3324	. 2 |- ( ph -> ( A \ C ) = ( B \ C ) )
3		difeq12d.2	. . 3 |- ( ph -> C = D )
4	3	difeq2d 3325	. 2 |- ( ph -> ( B \ C ) = ( B \ D ) )
5	2, 4	eqtrd 2264	1 |- ( ph -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   = wceq 1397   \ cdif 3197

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-dif 3202

This theorem is referenced by:  undifexmid  4283  exmidundif  4296  exmidundifim  4297