Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfunid | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4378. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 4374 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfvd 1831 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
6 | 3, 4, 5 | nfrexd 2989 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) |
7 | 2, 6 | nfabd 2771 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) |
8 | 1, 7 | nfcxfrd 2750 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2596 Ⅎwnfc 2738 ∃wrex 2897 ∪ cuni 4372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-uni 4373 |
This theorem is referenced by: dfnfc2 4390 dfnfc2OLD 4391 nfiotad 5771 |
Copyright terms: Public domain | W3C validator |