Theorem oveqdr 5905

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 5894	. 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 1353  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  grppropstrg  12900  grpsubpropdg  12979  crngpropd  13223  isringd  13225  ring1  13241  opprring  13254  opprringbg  13255  mulgass3  13259  rngidpropdg  13320  invrpropdg  13323  subrgpropd  13374