Theorem oveqdr 5925

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	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
oveqdr	⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

Proof of Theorem oveqdr

Step	Hyp	Ref	Expression
1		oveqdr.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2	1	oveqd 5914	. 2 ⊢ (𝜑 → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3	2	adantr 276	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  (class class class)co 5897

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900

This theorem is referenced by:  gsumpropd  12870  grppropstrg  12979  grpsubpropdg  13063  isrngd  13324  crngpropd  13410  isringd  13412  ring1  13428  opprrng  13444  opprrngbg  13445  opprring  13446  opprringbg  13447  opprsubgg  13451  mulgass3  13452  rngidpropdg  13513  invrpropdg  13516  subrngpropd  13580  subrgpropd  13612  sraring  13782  sralmod  13783  sralmod0g  13784  issubrgd  13785  rlmvnegg  13798  lidlrsppropdg  13828  crngridl  13861  zncrng  13957