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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 2895 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | cbviunf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
6 | 5 | nfcri 2895 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
7 | cbviunf.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
8 | 7 | eleq2d 2825 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
9 | 1, 2, 4, 6, 8 | cbvrexfw 3303 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
10 | 9 | abbii 2807 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
11 | df-iun 4998 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
12 | df-iun 4998 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
13 | 10, 11, 12 | 3eqtr4i 2773 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ∃wrex 3068 ∪ 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-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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: aciunf1lem 32679 |
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