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Mirrors > Home > ILE Home > Th. List > rzal | GIF version |
Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rzal | ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3410 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1 | necon2bi 2389 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
3 | 2 | pm2.21d 609 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 3 | ralrimiv 2536 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∅c0 3404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-v 2723 df-dif 3113 df-nul 3405 |
This theorem is referenced by: ralf0 3507 |
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