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Mirrors > Home > MPE Home > Th. List > vtocl3g | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3478 with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.) |
Ref | Expression |
---|---|
vtocl3g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl3g.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl3g.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
vtocl3g.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl3g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3419 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl3g.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 344 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒))) |
4 | vtocl3g.3 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
5 | 4 | imbi2d 344 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ V → 𝜒) ↔ (𝐴 ∈ V → 𝜃))) |
6 | vtocl3g.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | vtocl3g.4 | . . . . 5 ⊢ 𝜑 | |
8 | 6, 7 | vtoclg 3474 | . . . 4 ⊢ (𝐴 ∈ V → 𝜓) |
9 | 3, 5, 8 | vtocl2g 3479 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃)) |
10 | 1, 9 | mpan9 510 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃) |
11 | 10 | 3impb 1117 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 |
This theorem is referenced by: vtocl3ga 3486 |
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