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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 2897 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 | 
| 5 | cbviunf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 6 | 5 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 | 
| 7 | cbviunf.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 8 | 7 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 9 | 1, 2, 4, 6, 8 | cbvrexfw 3305 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 10 | 9 | abbii 2809 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | 
| 11 | df-iun 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
| 12 | df-iun 4993 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
| 13 | 10, 11, 12 | 3eqtr4i 2775 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∃wrex 3070 ∪ ciun 4991 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-iun 4993 | 
| This theorem is referenced by: aciunf1lem 32672 | 
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