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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 2883 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | cbviunf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
6 | 5 | nfcri 2883 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
7 | cbviunf.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
8 | 7 | eleq2d 2812 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
9 | 1, 2, 4, 6, 8 | cbvrexfw 3293 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
10 | 9 | abbii 2796 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
11 | df-iun 5003 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
12 | df-iun 5003 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
13 | 10, 11, 12 | 3eqtr4i 2764 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2703 Ⅎwnfc 2876 ∃wrex 3060 ∪ ciun 5001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-iun 5003 |
This theorem is referenced by: aciunf1lem 32579 |
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