![]() |
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 4919. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 4914 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfv 1912 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfvd 1913 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
6 | 3, 4, 5 | nfrexdw 3308 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) |
7 | 2, 6 | nfabdw 2925 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) |
8 | 1, 7 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2712 Ⅎwnfc 2888 ∃wrex 3068 ∪ cuni 4912 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-uni 4913 |
This theorem is referenced by: nfuni 4919 dfnfc2 4934 nfiotadw 6519 nfiotad 6521 |
Copyright terms: Public domain | W3C validator |