Theorem oveqdr 5953

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 5942	. 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 1364  (class class class)co 5925

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  gsumpropd  13094  grppropstrg  13221  grpsubpropdg  13306  isrngd  13585  crngpropd  13671  isringd  13673  ring1  13691  opprrng  13709  opprrngbg  13710  opprring  13711  opprringbg  13712  opprsubgg  13716  mulgass3  13717  rngidpropdg  13778  invrpropdg  13781  subrngpropd  13848  subrgpropd  13885  isdomn  13901  sraring  14081  sralmod  14082  sralmod0g  14083  issubrgd  14084  rlmvnegg  14097  lidlrsppropdg  14127  crngridl  14162  znzrh  14275  zncrng  14277