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Theorem oveqrspc2v 6572
 Description: Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
Hypothesis
Ref Expression
oveqrspc2v.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
Assertion
Ref Expression
oveqrspc2v ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝑦,𝑌   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem oveqrspc2v
StepHypRef Expression
1 oveqrspc2v.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
21ralrimivva 2954 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3 oveq1 6556 . . . 4 (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
4 oveq1 6556 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
53, 4eqeq12d 2625 . . 3 (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦)))
6 oveq2 6557 . . . 4 (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
7 oveq2 6557 . . . 4 (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
86, 7eqeq12d 2625 . . 3 (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
95, 8rspc2v 3293 . 2 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
102, 9mpan9 485 1 ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  grpidpropd  17084  gsumpropd2lem  17096  mndpropd  17139  grpsubpropd2  17344  cmnpropd  18025  ringpropd  18405  lmodprop2d  18748  lsspropd  18838  lmhmpropd  18894  lbspropd  18920  assapropd  19148  asclpropd  19167  psrplusgpropd  19427  phlpropd  19819
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