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Mirrors > Home > ILE Home > Th. List > vtoclb | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
vtoclb.1 | ⊢ 𝐴 ∈ V |
vtoclb.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
vtoclb.3 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
vtoclb.4 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
vtoclb | ⊢ (𝜒 ↔ 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclb.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | vtoclb.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
3 | vtoclb.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) | |
4 | 2, 3 | bibi12d 233 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
5 | vtoclb.4 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
6 | 1, 4, 5 | vtocl 2653 | 1 ⊢ (𝜒 ↔ 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 |
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-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: alexeq 2721 sbss 3349 |
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