Theorem difeq12d 3278

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 3276	. 2 |- ( ph -> ( A \ C ) = ( B \ C ) )
3		difeq12d.2	. . 3 |- ( ph -> C = D )
4	3	difeq2d 3277	. 2 |- ( ph -> ( B \ C ) = ( B \ D ) )
5	2, 4	eqtrd 2226	1 |- ( ph -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   = wceq 1364   \ cdif 3150

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-dif 3155

This theorem is referenced by:  undifexmid  4222  exmidundif  4235  exmidundifim  4236