Theorem oveqdr 5974

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 5963	. 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 1373  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  gsumpropd  13257  grppropstrg  13384  grpsubpropdg  13469  isrngd  13748  crngpropd  13834  isringd  13836  ring1  13854  opprrng  13872  opprrngbg  13873  opprring  13874  opprringbg  13875  opprsubgg  13879  mulgass3  13880  rngidpropdg  13941  invrpropdg  13944  subrngpropd  14011  subrgpropd  14048  isdomn  14064  sraring  14244  sralmod  14245  sralmod0g  14246  issubrgd  14247  rlmvnegg  14260  lidlrsppropdg  14290  crngridl  14325  znzrh  14438  zncrng  14440