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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.) Add disjoint variable condition to avoid ax-13 2374. See nfiundg 48685 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) |
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 5021 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfv 1913 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiund.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfiund.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
5 | nfiund.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵) | |
6 | 5 | nfcrd 2897 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfrexdw 3311 | . . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabdw 2928 | . 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2903 | 1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1781 ∈ wcel 2103 {cab 2711 Ⅎwnfc 2888 ∃wrex 3072 ∪ ciun 5019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-iun 5021 |
This theorem is referenced by: (None) |
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