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| Mirrors > Home > MPE Home > Th. List > Mathboxes > r19.41dv | Structured version Visualization version GIF version | ||
| Description: A complex deduction form of r19.41v 3189. (Contributed by Zhi Wang, 6-Sep-2024.) |
| Ref | Expression |
|---|---|
| r19.41dv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
| Ref | Expression |
|---|---|
| r19.41dv | ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.41dv.1 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
| 2 | 1 | anim1i 615 | . 2 ⊢ ((𝜑 ∧ 𝜒) → (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜒)) |
| 3 | r19.41v 3189 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜒)) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-rex 3071 |
| This theorem is referenced by: opnneilv 48806 |
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