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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3rspcedvd | Structured version Visualization version GIF version |
Description: Triple application of rspcedvd 3623. (Contributed by Steven Nguyen, 27-Feb-2023.) |
Ref | Expression |
---|---|
3rspcedvd.a | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
3rspcedvd.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
3rspcedvd.c | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
3rspcedvd.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
3rspcedvd.2 | ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) |
3rspcedvd.3 | ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) |
3rspcedvd.4 | ⊢ (𝜑 → 𝜏) |
Ref | Expression |
---|---|
3rspcedvd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3rspcedvd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | 3rspcedvd.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | 2rexbidv 3297 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓 ↔ ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒)) |
4 | 3rspcedvd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 3rspcedvd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) | |
6 | 5 | rexbidv 3294 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (∃𝑧 ∈ 𝐷 𝜒 ↔ ∃𝑧 ∈ 𝐷 𝜃)) |
7 | 3rspcedvd.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
8 | 3rspcedvd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) | |
9 | 3rspcedvd.4 | . . . 4 ⊢ (𝜑 → 𝜏) | |
10 | 7, 8, 9 | rspcedvd 3623 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 𝜃) |
11 | 4, 6, 10 | rspcedvd 3623 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒) |
12 | 1, 3, 11 | rspcedvd 3623 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-ral 3140 df-rex 3141 |
This theorem is referenced by: dffltz 39149 |
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