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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviunf | Structured version Visualization version GIF version |
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
cbviunf.x | ⊢ Ⅎ𝑥𝐴 |
cbviunf.y | ⊢ Ⅎ𝑦𝐴 |
cbviunf.1 | ⊢ Ⅎ𝑦𝐵 |
cbviunf.2 | ⊢ Ⅎ𝑥𝐶 |
cbviunf.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviunf | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviunf.x | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | cbviunf.y | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | cbviunf.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2929 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | cbviunf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
6 | 5 | nfcri 2929 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
7 | cbviunf.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
8 | 7 | eleq2d 2845 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
9 | 1, 2, 4, 6, 8 | cbvrexf 3362 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
10 | 9 | abbii 2908 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
11 | df-iun 4757 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
12 | df-iun 4757 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
13 | 10, 11, 12 | 3eqtr4i 2812 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {cab 2763 Ⅎwnfc 2919 ∃wrex 3091 ∪ ciun 4755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-iun 4757 |
This theorem is referenced by: aciunf1lem 30031 |
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