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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 1421 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfcvd 2179 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfvd 1422 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
7 | 1, 2, 3, 4, 5, 6 | vtocldf 2605 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: funfvima3 5392 frec2uzzd 9186 frec2uzuzd 9188 |
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