Theorem oveqdr 6056

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 6045	. 2 ⊢ (𝜑 → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3	2	adantr 276	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  gsumpropd  13555  grppropstrg  13682  grpsubpropdg  13767  isrngd  14047  crngpropd  14133  isringd  14135  ring1  14153  opprrng  14171  opprrngbg  14172  opprring  14173  opprringbg  14174  opprsubgg  14178  mulgass3  14179  rngidpropdg  14241  invrpropdg  14244  subrngpropd  14311  subrgpropd  14348  isdomn  14365  sraring  14545  sralmod  14546  sralmod0g  14547  issubrgd  14548  rlmvnegg  14561  lidlrsppropdg  14591  crngridl  14626  znzrh  14739  zncrng  14741