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| Description: Deduction version of nfuni 4913. (Contributed by NM, 18-Feb-2013.) | 
| Ref | Expression | 
|---|---|
| nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| Ref | Expression | 
|---|---|
| nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfuni2 4908 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
| 2 | nfv 1913 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
| 4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | nfvd 1914 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
| 6 | 3, 4, 5 | nfrexdw 3309 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) | 
| 7 | 2, 6 | nfabdw 2926 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) | 
| 8 | 1, 7 | nfcxfrd 2903 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 {cab 2713 Ⅎwnfc 2889 ∃wrex 3069 ∪ cuni 4906 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-uni 4907 | 
| This theorem is referenced by: nfuni 4913 dfnfc2 4928 nfiotadw 6516 nfiotad 6518 | 
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