Theorem oveqdr 6049

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 6038	. 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 1397  (class class class)co 6021

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6024

This theorem is referenced by:  gsumpropd  13496  grppropstrg  13623  grpsubpropdg  13708  isrngd  13988  crngpropd  14074  isringd  14076  ring1  14094  opprrng  14112  opprrngbg  14113  opprring  14114  opprringbg  14115  opprsubgg  14119  mulgass3  14120  rngidpropdg  14182  invrpropdg  14185  subrngpropd  14252  subrgpropd  14289  isdomn  14305  sraring  14485  sralmod  14486  sralmod0g  14487  issubrgd  14488  rlmvnegg  14501  lidlrsppropdg  14531  crngridl  14566  znzrh  14679  zncrng  14681