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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 3157 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
8 | rspcdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
9 | 8, 2 | rspcedv 3602 | . 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 1534 ∈ wcel 2099 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 |
This theorem is referenced by: fusgr2wsp2nb 30143 |
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