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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspceb2dv | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.) |
Ref | Expression |
---|---|
rspceb2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒)) |
rspceb2dv.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵) |
rspceb2dv.3 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
rspceb2dv.4 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
rspceb2dv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspceb2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒)) | |
2 | 1 | rexlimdva 3212 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
3 | rspceb2dv.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵) | |
4 | rspceb2dv.3 | . . . 4 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
5 | rspceb2dv.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) | |
6 | 5 | rspcev 3561 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜃) → ∃𝑥 ∈ 𝐵 𝜓) |
7 | 3, 4, 6 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓) |
8 | 7 | ex 413 | . 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
9 | 2, 8 | impbid 211 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 |
This theorem is referenced by: ipolubdm 46230 ipoglbdm 46233 |
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