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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 3186. (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 3186 | . 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 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-rex 3068 |
This theorem is referenced by: opnneilv 48704 |
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