Theorem oveqdr 6078

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 6067	. 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 6050

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  gsumpropd  13605  grppropstrg  13732  grpsubpropdg  13817  isrngd  14097  crngpropd  14183  isringd  14185  ring1  14203  opprrng  14221  opprrngbg  14222  opprring  14223  opprringbg  14224  opprsubgg  14228  mulgass3  14229  rngidpropdg  14291  invrpropdg  14294  subrngpropd  14361  subrgpropd  14398  isdomn  14415  sraring  14597  sralmod  14598  sralmod0g  14599  issubrgd  14600  rlmvnegg  14613  lidlrsppropdg  14643  crngridl  14678  znzrh  14791  zncrng  14793