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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfiund | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) |
Ref | Expression |
---|---|
nfiund.1 | ⊢ Ⅎ𝑥𝜑 |
nfiund.2 | ⊢ (𝜑 → Ⅎ𝑦𝐴) |
nfiund.3 | ⊢ (𝜑 → Ⅎ𝑦𝐵) |
Ref | Expression |
---|---|
nfiund | ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4522 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfv 1843 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiund.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfiund.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
5 | nfiund.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵) | |
6 | 5 | nfcrd 2771 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfrexd 3006 | . . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2785 | . 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2763 | 1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1708 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 ∃wrex 2913 ∪ ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-iun 4522 |
This theorem is referenced by: (None) |
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