Theorem oveqdr 5946

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 5935	. 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 5918

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 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  gsumpropd  12975  grppropstrg  13091  grpsubpropdg  13176  isrngd  13449  crngpropd  13535  isringd  13537  ring1  13555  opprrng  13573  opprrngbg  13574  opprring  13575  opprringbg  13576  opprsubgg  13580  mulgass3  13581  rngidpropdg  13642  invrpropdg  13645  subrngpropd  13712  subrgpropd  13749  isdomn  13765  sraring  13945  sralmod  13946  sralmod0g  13947  issubrgd  13948  rlmvnegg  13961  lidlrsppropdg  13991  crngridl  14026  znzrh  14131  zncrng  14133