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Mirrors > Home > ILE Home > Th. List > iuneq2d | GIF version |
Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.) |
Ref | Expression |
---|---|
iuneq2d.2 | ⊢ (φ → B = 𝐶) |
Ref | Expression |
---|---|
iuneq2d | ⊢ (φ → ∪ x ∈ A B = ∪ x ∈ A 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2d.2 | . . 3 ⊢ (φ → B = 𝐶) | |
2 | 1 | adantr 261 | . 2 ⊢ ((φ ∧ x ∈ A) → B = 𝐶) |
3 | 2 | iuneq2dv 3669 | 1 ⊢ (φ → ∪ x ∈ A B = ∪ x ∈ A 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 ∪ ciun 3648 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-in 2918 df-ss 2925 df-iun 3650 |
This theorem is referenced by: rdgeq1 5898 |
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