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Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq12f | Structured version Visualization version GIF version |
Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
iuneq12f.1 | ⊢ Ⅎ𝑥𝐴 |
iuneq12f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
iuneq12f | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2 4473 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐴 𝐷) | |
2 | iuneq12f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | iuneq12f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | iuneq2f 33133 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷) |
5 | 1, 4 | sylan9eqr 2666 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnfc 2738 ∀wral 2896 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iun 4457 |
This theorem is referenced by: (None) |
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