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Mirrors > Home > MPE Home > Th. List > nfunid | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4837. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 4832 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfvd 1907 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
6 | 3, 4, 5 | nfrexd 3304 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) |
7 | 2, 6 | nfabdw 2997 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) |
8 | 1, 7 | nfcxfrd 2973 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2796 Ⅎwnfc 2958 ∃wrex 3136 ∪ 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: nfuni 4837 dfnfc2 4848 nfiotadw 6310 nfiotad 6312 |
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