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Mirrors > Home > MPE Home > Th. List > rspcebdv | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.) |
Ref | Expression |
---|---|
rspcdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
rspcebdv.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴) |
Ref | Expression |
---|---|
rspcebdv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcebdv.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴) | |
2 | rspcdv.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | syldan 590 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 ↔ 𝜒)) |
4 | 3 | biimpd 228 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜒)) |
5 | 4 | expcom 413 | . . . 4 ⊢ (𝜓 → (𝜑 → (𝜓 → 𝜒))) |
6 | 5 | pm2.43b 55 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
7 | 6 | rexlimdvw 3152 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
8 | rspcdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
9 | 8, 2 | rspcedv 3597 | . 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
10 | 7, 9 | impbid 211 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 |
This theorem is referenced by: fusgr2wsp2nb 30081 |
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