Theorem oveqdr 5947

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 5936	. 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 5919

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922

This theorem is referenced by:  gsumpropd  12978  grppropstrg  13094  grpsubpropdg  13179  isrngd  13452  crngpropd  13538  isringd  13540  ring1  13558  opprrng  13576  opprrngbg  13577  opprring  13578  opprringbg  13579  opprsubgg  13583  mulgass3  13584  rngidpropdg  13645  invrpropdg  13648  subrngpropd  13715  subrgpropd  13752  isdomn  13768  sraring  13948  sralmod  13949  sralmod0g  13950  issubrgd  13951  rlmvnegg  13964  lidlrsppropdg  13994  crngridl  14029  znzrh  14142  zncrng  14144