Theorem oveqrspc2v 6102

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 2632	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		oveq1 6082	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
4		oveq1 6082	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
5	3, 4	eqeq12d 2253	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦)))
6		oveq2 6083	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
7		oveq2 6083	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
8	6, 7	eqeq12d 2253	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌)))
9	5, 8	rspc2v 2943	. 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 1402   ∈ wcel 2209  ∀wral 2528  (class class class)co 6075

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  grpidpropdg  13671  sgrppropd  13705  mndpropd  13730  grpsubpropd2  13887  cmnpropd  14075  rngpropd  14229  ringpropd  14316  lmodprop2d  14657  lsspropdg  14740