Theorem oveqdr 5893

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 5882	. 2 ⊢ (𝜑 → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3	2	adantr 276	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  (class class class)co 5865

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  grppropstrg  12755  grpsubpropdg  12833  crngpropd  13012  isringd  13014  ring1  13030  opprring  13042  opprringbg  13043  mulgass3  13047