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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunidALT | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4837. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfunidALT.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunidALT | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfunidALT.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
2 | abidnf 3691 | . . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴) | |
3 | 2 | unieqd 4840 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴) |
4 | nfaba1 2983 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | |
5 | 4 | nfuni 4837 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
6 | 1, 3, 5 | nfded 35983 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-uni 4831 |
This theorem is referenced by: (None) |
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