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Mathbox for Giovanni Mascellani |
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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 4972 | . 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 36604 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷) |
5 | 1, 4 | sylan9eqr 2798 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnfc 2886 ∀wral 3063 ∪ ciun 4953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ral 3064 df-rex 3073 df-v 3446 df-in 3916 df-ss 3926 df-iun 4955 |
This theorem is referenced by: (None) |
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