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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunidALT2 | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4922. (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 2900 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | |
2 | 1 | nfuni 4922 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
3 | nfunidALT2.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfnfc1 2895 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
5 | abidnf 3696 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴) | |
6 | 5 | unieqd 4928 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴) |
7 | 4, 6 | nfceqdf 2887 | . . 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 232 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 ∈ wcel 2099 {cab 2703 Ⅎwnfc 2876 ∪ cuni 4915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-v 3464 df-ss 3964 df-uni 4916 |
This theorem is referenced by: (None) |
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