Theorem oveqan12rd 5885

Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Hypotheses

Ref	Expression
oveq1d.1	|- ( ph -> A = B )
opreqan12i.2	|- ( ps -> C = D )

Assertion

Ref	Expression
oveqan12rd	|- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) )

Proof of Theorem oveqan12rd

Step	Hyp	Ref	Expression
1		oveq1d.1	. . 3 |- ( ph -> A = B )
2		opreqan12i.2	. . 3 |- ( ps -> C = D )
3	1, 2	oveqan12d 5884	. 2 |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) )
4	3	ancoms 268	1 |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353  (class class class)co 5865

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  mulresr  7812  recdivap  8648  divgcdcoprm0  12068  ismhm  12716  dvid  13733