Theorem oveqdr 6113

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 6102	. 2 |- ( ph -> ( x F y ) = ( x G y ) )
3	2	adantr 276	1 |- ( ( ph /\ ps ) -> ( x F y ) = ( x G y ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  grppropstrg  13826  grpsubpropdg  13911  isrngd  14254  crngpropd  14346  isringd  14348  ring1  14366  opprrng  14384  opprrngbg  14385  opprring  14386  opprringbg  14387  opprsubgg  14392  mulgass3  14393  rngidpropdg  14455  invrpropdg  14458  subrngpropd  14526  subrgpropd  14563  isdomn  14580  aprprop  14603  sraring  14788  sralmod  14789  sralmod0g  14790  issubrgd  14791  rlmvnegg  14804  lidlrsppropdg  14834  crngridl  14869  znzrh  14980  zncrng  14982