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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3lem7 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3rlem8 45693 and iscnrm3llem2 45696 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm3lem7.1 | ⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃)) |
iscnrm3lem7.2 | ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏)) |
iscnrm3lem7.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏)) |
Ref | Expression |
---|---|
iscnrm3lem7 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscnrm3lem7.3 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏)) | |
2 | iscnrm3lem7.1 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃)) | |
3 | iscnrm3lem7.2 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏)) | |
4 | 2, 3 | rspc2ev 3556 | . . 3 ⊢ ((𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏) → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒) |
5 | 1, 4 | syl 17 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒) |
6 | 5 | iscnrm3lem6 45684 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∃wrex 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1087 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ral 3076 df-rex 3077 |
This theorem is referenced by: iscnrm3rlem8 45693 iscnrm3llem2 45696 |
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