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Mirrors > Home > MPE Home > Th. List > vtoclb | Structured version Visualization version 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 334 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
5 | vtoclb.4 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
6 | 1, 4, 5 | vtocl 3232 | 1 ⊢ (𝜒 ↔ 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-tru 1478 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: sbss 4034 bnj609 30241 |
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