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| Mirrors > Home > MPE Home > Th. List > ab0ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of ab0 4380, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ab0ALT | ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfab1 2907 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
| 2 | 1 | eq0f 4347 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑}) |
| 3 | abid 2718 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 4 | 3 | notbii 320 | . . 3 ⊢ (¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑) |
| 5 | 4 | albii 1819 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ¬ 𝜑) |
| 6 | 2, 5 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: (None) |
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