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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimdv3d | Structured version Visualization version GIF version |
Description: An extended version of rexlimdvv 3212 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.) |
Ref | Expression |
---|---|
rexlimdv3d.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlimdv3d | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdv3d.1 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒))) | |
2 | 1 | 3expd 1355 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐶 → (𝜓 → 𝜒))))) |
3 | 2 | imp4d 428 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → (𝜓 → 𝜒))) |
4 | 3 | expdimp 456 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒))) |
5 | 4 | rexlimdvv 3212 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒)) |
6 | 5 | rexlimdva 3203 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 ∃wrex 3062 |
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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-ex 1788 df-ral 3066 df-rex 3067 |
This theorem is referenced by: 3cubes 40215 |
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