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Mirrors > Home > MPE Home > Th. List > ralf0 | Structured version Visualization version GIF version |
Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
ralf0.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
ralf0 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralf0.1 | . . . 4 ⊢ ¬ 𝜑 | |
2 | mtt 353 | . . . 4 ⊢ (¬ 𝜑 → (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 3 | albii 1737 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
5 | eq0 3888 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
6 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | 4, 5, 6 | 3bitr4ri 292 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 |
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-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: uvtx01vtx 26020 rusgra0edg 26482 |
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