Theorem oveqdr 6041

Description: Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Hypothesis

Ref	Expression
oveqdr.1	|- ( ph -> F = G )

Assertion

Ref	Expression
oveqdr	|- ( ( ph /\ ps ) -> ( x F y ) = ( x G y ) )

Proof of Theorem oveqdr

Step	Hyp	Ref	Expression
1		oveqdr.1	. . 3 |- ( ph -> F = G )
2	1	oveqd 6030	. 2 |- ( ph -> ( x F y ) = ( x G y ) )
3	2	adantr 276	1 |- ( ( ph /\ ps ) -> ( x F y ) = ( x G y ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395  (class class class)co 6013

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by:  gsumpropd  13465  grppropstrg  13592  grpsubpropdg  13677  isrngd  13956  crngpropd  14042  isringd  14044  ring1  14062  opprrng  14080  opprrngbg  14081  opprring  14082  opprringbg  14083  opprsubgg  14087  mulgass3  14088  rngidpropdg  14150  invrpropdg  14153  subrngpropd  14220  subrgpropd  14257  isdomn  14273  sraring  14453  sralmod  14454  sralmod0g  14455  issubrgd  14456  rlmvnegg  14469  lidlrsppropdg  14499  crngridl  14534  znzrh  14647  zncrng  14649