![]() |
Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
|
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 2917 | . 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
5 | eqidd 2736 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
6 | 3, 1, 2, 4, 5 | iuneq12df 5023 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2888 ∪ ciun 4996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-iun 4998 |
This theorem is referenced by: iuneq12f 38150 |
Copyright terms: Public domain | W3C validator |