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Mirrors > Home > ILE Home > Th. List > vtoclgf | GIF version |
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtoclgf.1 | ⊢ Ⅎ𝑥𝐴 |
vtoclgf.2 | ⊢ Ⅎ𝑥𝜓 |
vtoclgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclgf.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtoclgf | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 2693 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | vtoclgf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | vtoclgf.4 | . . . . 5 ⊢ 𝜑 | |
6 | vtoclgf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | mpbii 147 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
8 | 4, 7 | exlimi 1573 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
9 | 3, 8 | sylbi 120 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
10 | 1, 9 | 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 Ⅎwnfc 2268 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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 |
This theorem is referenced by: vtoclg 2746 vtocl2gf 2748 vtocl3gf 2749 vtoclgaf 2751 ceqsexg 2813 elabgf 2826 mob 2866 opeliunxp2 4679 fvmptss2 5496 |
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