Theorem difeq12d 3190

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 3188	. 2 |- ( ph -> ( A \ C ) = ( B \ C ) )
3		difeq12d.2	. . 3 |- ( ph -> C = D )
4	3	difeq2d 3189	. 2 |- ( ph -> ( B \ C ) = ( B \ D ) )
5	2, 4	eqtrd 2170	1 |- ( ph -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   = wceq 1331   \ cdif 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-dif 3068

This theorem is referenced by:  undifexmid  4112  exmidundif  4124  exmidundifim  4125