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Mirrors > Home > MPE Home > Th. List > cbviung | 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. Usage of this theorem is discouraged because it depends on ax-13 2365. See cbviun 5032 for a version with more disjoint variable conditions, but not requiring ax-13 2365. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbviung.1 | ⊢ Ⅎ𝑦𝐵 |
cbviung.2 | ⊢ Ⅎ𝑥𝐶 |
cbviung.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviung | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviung.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
2 | 1 | nfcri 2884 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
3 | cbviung.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2884 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
5 | cbviung.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2813 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
7 | 2, 4, 6 | cbvrex 3353 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
8 | 7 | abbii 2796 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
9 | df-iun 4992 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
10 | df-iun 4992 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
11 | 8, 9, 10 | 3eqtr4i 2764 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cab 2703 Ⅎwnfc 2877 ∃wrex 3064 ∪ ciun 4990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2365 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-iun 4992 |
This theorem is referenced by: cbviunvg 5038 |
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