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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3rspcedvdw | Structured version Visualization version GIF version |
Description: Triple application of rspcedvdw 39942. (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
3rspcedvdw.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
3rspcedvdw.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) |
3rspcedvdw.3 | ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) |
3rspcedvdw.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
3rspcedvdw.b | ⊢ (𝜑 → 𝐵 ∈ 𝑌) |
3rspcedvdw.c | ⊢ (𝜑 → 𝐶 ∈ 𝑍) |
3rspcedvdw.4 | ⊢ (𝜑 → 𝜏) |
Ref | Expression |
---|---|
3rspcedvdw | ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3rspcedvdw.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | 3rspcedvdw.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑌) | |
3 | 3rspcedvdw.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑍) | |
4 | 3rspcedvdw.4 | . 2 ⊢ (𝜑 → 𝜏) | |
5 | 3rspcedvdw.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
6 | 3rspcedvdw.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
7 | 3rspcedvdw.3 | . . 3 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) | |
8 | 5, 6, 7 | rspc3ev 3566 | . 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝜏) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓) |
9 | 1, 2, 3, 4, 8 | syl31anc 1375 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3065 |
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 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 |
This theorem is referenced by: dffltz 40221 nna4b4nsq 40247 |
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