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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunidALT | Structured version Visualization version GIF version | ||
| Description: Deduction version of nfuni 4914. (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 3708 | . . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴) | |
| 3 | 2 | unieqd 4920 | . 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴) |
| 4 | nfaba1 2913 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | |
| 5 | 4 | nfuni 4914 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
| 6 | 1, 3, 5 | nfded 38968 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∪ cuni 4907 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 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-v 3482 df-ss 3968 df-uni 4908 |
| This theorem is referenced by: (None) |
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