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Theorem oveqrspc2v 7385
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 3198 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3 oveq1 7365 . . . 4 (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
4 oveq1 7365 . . . 4 (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
53, 4eqeq12d 2753 . . 3 (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦)))
6 oveq2 7366 . . . 4 (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
7 oveq2 7366 . . . 4 (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
86, 7eqeq12d 2753 . . 3 (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
95, 8rspc2v 3591 . 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 3065  (class class class)co 7358
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  grpidpropd  18518  gsumpropd2lem  18535  mndpropd  18582  grpsubpropd2  18854  cmnpropd  19574  ringpropd  20007  lmodprop2d  20387  lsspropd  20481  lmhmpropd  20537  lbspropd  20563  phlpropd  21062  assapropd  21278  asclpropd  21303  psrplusgpropd  21610  lindfpropd  32172
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