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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3rspcedvd | Structured version Visualization version GIF version |
Description: Triple application of rspcedvd 3564. (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 3231 | . 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 3228 | . . 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 3564 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 𝜃) |
11 | 4, 6, 10 | rspcedvd 3564 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜒) |
12 | 1, 3, 11 | rspcedvd 3564 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 |
This theorem is referenced by: (None) |
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