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Mirrors > Home > MPE Home > Th. List > oveqrspc2v | Structured version Visualization version GIF version |
Description: Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.) |
Ref | Expression |
---|---|
oveqrspc2v.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
Ref | Expression |
---|---|
oveqrspc2v | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveqrspc2v.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) | |
2 | 1 | ralrimivva 3198 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
3 | oveq1 7365 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦)) | |
4 | oveq1 7365 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦)) | |
5 | 3, 4 | eqeq12d 2753 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝑋𝐹𝑦) = (𝑋𝐺𝑦))) |
6 | oveq2 7366 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌)) | |
7 | oveq2 7366 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌)) | |
8 | 6, 7 | eqeq12d 2753 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) = (𝑋𝐺𝑦) ↔ (𝑋𝐹𝑌) = (𝑋𝐺𝑌))) |
9 | 5, 8 | rspc2v 3591 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))) |
10 | 2, 9 | mpan9 508 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (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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