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Mirrors > Home > ILE Home > Th. List > vtoclg1f | GIF version |
Description: Version of vtoclgf 2795 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1506 and ax-13 2150. (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 2748 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2743 | . . 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 148 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
7 | 3, 6 | exlimi 1594 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
8 | 2, 7 | sylbi 121 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
9 | 1, 8 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 Ⅎwnf 1460 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 |
This theorem is referenced by: opeliunxp2f 6236 summodclem2a 11382 fprodsplit1f 11635 |
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