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Theorem oveqrspc2v 7436
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 3201 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3 oveq1 7416 . . . 4 (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
4 oveq1 7416 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
53, 4eqeq12d 2749 . . 3 (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦)))
6 oveq2 7417 . . . 4 (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
7 oveq2 7417 . . . 4 (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
86, 7eqeq12d 2749 . . 3 (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
95, 8rspc2v 3623 . 2 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
102, 9mpan9 508 1 ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  (class class class)co 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  grpidpropd  18581  gsumpropd2lem  18598  sgrppropd  18622  mndpropd  18650  grpsubpropd2  18929  cmnpropd  19659  ringpropd  20102  lmodprop2d  20534  lsspropd  20628  lmhmpropd  20684  lbspropd  20710  phlpropd  21208  assapropd  21426  asclpropd  21451  psrplusgpropd  21758  lindfpropd  32498  rngpropd  46673
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