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| Mirrors > Home > MPE Home > Th. List > rexlimiva | Structured version Visualization version GIF version | ||
| Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.) Shorten dependent theorems. (Revised by Wolf lammen, 23-Dec-2024.) |
| Ref | Expression |
|---|---|
| rexlimiva.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
| Ref | Expression |
|---|---|
| rexlimiva | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | rexlimiva.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) | |
| 3 | 2 | exlimiv 1930 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
| 4 | 1, 3 | sylbi 217 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
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