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Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq2f | Structured version Visualization version GIF version |
Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Ref | Expression |
---|---|
iuneq2f.1 | ⊢ Ⅎ𝑥𝐴 |
iuneq2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
iuneq2f | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | iuneq2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2990 | . 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
5 | eqidd 2821 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
6 | 3, 1, 2, 4, 5 | iuneq12df 4938 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Ⅎwnfc 2960 ∪ ciun 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 df-iun 4914 |
This theorem is referenced by: iuneq12f 35474 |
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