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Mirrors > Home > ILE Home > Th. List > vtoclg1f | GIF version |
Description: Version of vtoclgf 2744 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1484 and ax-13 1491. (Contributed by BJ, 1-May-2019.) |
Ref | Expression |
---|---|
vtoclg1f.nf | ⊢ Ⅎ𝑥𝜓 |
vtoclg1f.maj | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclg1f.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtoclg1f | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2692 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | vtoclg1f.nf | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | vtoclg1f.min | . . . . 5 ⊢ 𝜑 | |
5 | vtoclg1f.maj | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | mpbii 147 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
7 | 3, 6 | exlimi 1573 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
8 | 2, 7 | sylbi 120 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
9 | 1, 8 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 |
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-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 |
This theorem is referenced by: opeliunxp2f 6135 summodclem2a 11150 |
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