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Mirrors > Home > MPE Home > Th. List > nfunid | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4838. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 4833 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfvd 1912 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
6 | 3, 4, 5 | nfrexd 3307 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) |
7 | 2, 6 | nfabdw 3000 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) |
8 | 1, 7 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2799 Ⅎwnfc 2961 ∃wrex 3139 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-uni 4832 |
This theorem is referenced by: nfuni 4838 dfnfc2 4849 nfiotadw 6311 nfiotad 6313 |
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