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Mirrors > Home > MPE Home > Th. List > rspcedvd | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 3552. (Contributed by AV, 27-Nov-2019.) |
Ref | Expression |
---|---|
rspcedvd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcedvd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
rspcedvd.3 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
rspcedvd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcedvd.3 | . 2 ⊢ (𝜑 → 𝜒) | |
2 | rspcedvd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | rspcedvd.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | rspcedv 3552 | . 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
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