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Mirrors > Home > ILE Home > Th. List > vtoclef | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
vtoclef.1 | ⊢ Ⅎ𝑥𝜑 |
vtoclef.2 | ⊢ 𝐴 ∈ V |
vtoclef.3 | ⊢ (𝑥 = 𝐴 → 𝜑) |
Ref | Expression |
---|---|
vtoclef | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclef.2 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 2694 | . 2 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtoclef.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | vtoclef.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
5 | 3, 4 | exlimi 1573 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: nn0ind-raph 9168 |
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