Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunidALT2 | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4847. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfunidALT2.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunidALT2 | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfaba1 2988 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | |
2 | 1 | nfuni 4847 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
3 | nfunidALT2.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfnfc1 2982 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
5 | abidnf 3696 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴) | |
6 | 5 | unieqd 4854 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴) |
7 | 4, 6 | nfceqdf 2974 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} ↔ Ⅎ𝑥∪ 𝐴)) |
8 | 3, 7 | syl 17 | . 2 ⊢ (𝜑 → (Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} ↔ Ⅎ𝑥∪ 𝐴)) |
9 | 2, 8 | mpbii 235 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 {cab 2801 Ⅎwnfc 2963 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |