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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 234 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
5 | vtoclb.4 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
6 | 1, 4, 5 | vtocl 2714 | 1 ⊢ (𝜒 ↔ 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 Vcvv 2660 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 |
This theorem is referenced by: alexeq 2785 sbss 3441 |
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