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Mirrors > Home > ILE Home > Th. List > vtocld | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
vtocld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vtocld.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
vtocld.3 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
vtocld | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | vtocld.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | vtocld.3 | . 2 ⊢ (𝜑 → 𝜓) | |
4 | nfv 1461 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfcvd 2220 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfvd 1462 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
7 | 1, 2, 3, 4, 5, 6 | vtocldf 2650 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 |
This theorem is referenced by: funfvima3 5413 frec2uzzd 9402 frec2uzuzd 9404 |
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