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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfunidALT2 | Structured version Visualization version GIF version |
Description: Deduction version of nfuni 4378. (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 2756 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | |
2 | 1 | nfuni 4378 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
3 | nfunidALT2.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfnfc1 2754 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
5 | abidnf 3342 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴) | |
6 | 5 | unieqd 4382 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴) |
7 | 4, 6 | nfceqdf 2747 | . . 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 222 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 ∪ 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: (None) |
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