Theorem oveqrspc2v 5945

Description: Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)

Hypothesis

Ref	Expression
oveqrspc2v.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))

Assertion

Ref	Expression
oveqrspc2v	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (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

Step	Hyp	Ref	Expression
1		oveqrspc2v.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
2	1	ralrimivva 2576	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		oveq1 5925	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
4		oveq1 5925	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
5	3, 4	eqeq12d 2208	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦)))
6		oveq2 5926	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
7		oveq2 5926	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
8	6, 7	eqeq12d 2208	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
9	5, 8	rspc2v 2877	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
10	2, 9	mpan9 281	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  grpidpropdg  12957  sgrppropd  12996  mndpropd  13021  grpsubpropd2  13177  cmnpropd  13365  rngpropd  13451  ringpropd  13534  lmodprop2d  13844  lsspropdg  13927