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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 1539 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfcvd 2333 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfvd 1540 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
7 | 1, 2, 3, 4, 5, 6 | vtocldf 2803 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: funfvima3 5771 isbth 6996 frec2uzuzd 10433 setscomd 12553 |
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