Theorem oveqdr 6113

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 6102	. 2 ⊢ (𝜑 → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3	2	adantr 276	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  grppropstrg  13873  grpsubpropdg  13958  isrngd  14301  crngpropd  14393  isringd  14395  ring1  14413  opprrng  14431  opprrngbg  14432  opprring  14433  opprringbg  14434  opprsubgg  14439  mulgass3  14440  rngidpropdg  14502  invrpropdg  14505  subrngpropd  14573  subrgpropd  14610  isdomn  14627  aprprop  14650  sraring  14835  sralmod  14836  sralmod0g  14837  issubrgd  14838  rlmvnegg  14851  lidlrsppropdg  14881  crngridl  14916  znzrh  15027  zncrng  15029