Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralbinrald | Structured version Visualization version GIF version |
Description: Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.) |
Ref | Expression |
---|---|
ralbinrald.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
ralbinrald.2 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋) |
ralbinrald.3 | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
ralbinrald | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbinrald.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
2 | ralbinrald.3 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃)) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝜓 ↔ 𝜃)) |
4 | 1, 3 | rspcdv 3552 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜃)) |
5 | ralbinrald.2 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋) | |
6 | 2 | bicomd 222 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝜃 ↔ 𝜓)) |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝜃 ↔ 𝜓)) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 ↔ 𝜓)) |
9 | 8 | biimpd 228 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 → 𝜓)) |
10 | 9 | ralrimdva 3115 | . 2 ⊢ (𝜑 → (𝜃 → ∀𝑥 ∈ 𝐴 𝜓)) |
11 | 4, 10 | impbid 211 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 |
This theorem is referenced by: dfdfat2 44588 |
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