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Mirrors > Home > MPE Home > Th. List > ab0ALT | Structured version Visualization version GIF version |
Description: Alternate proof of ab0 4275, 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 2899 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | 1 | eq0f 4241 | . 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑}) |
3 | abid 2718 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
4 | 3 | notbii 323 | . . 3 ⊢ (¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑) |
5 | 4 | albii 1827 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ¬ 𝜑) |
6 | 2, 5 | bitri 278 | 1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2112 {cab 2714 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-dif 3856 df-nul 4224 |
This theorem is referenced by: (None) |
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