Theorem oveqdr 6080

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 6069	. 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 1398  (class class class)co 6052

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by:  gsumpropd  13622  grppropstrg  13749  grpsubpropdg  13834  isrngd  14114  crngpropd  14200  isringd  14202  ring1  14220  opprrng  14238  opprrngbg  14239  opprring  14240  opprringbg  14241  opprsubgg  14245  mulgass3  14246  rngidpropdg  14308  invrpropdg  14311  subrngpropd  14378  subrgpropd  14415  isdomn  14432  sraring  14614  sralmod  14615  sralmod0g  14616  issubrgd  14617  rlmvnegg  14630  lidlrsppropdg  14660  crngridl  14695  znzrh  14808  zncrng  14810