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| Mirrors > Home > MPE Home > Th. List > elabd2 | Structured version Visualization version GIF version | ||
| Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab 3679. (Contributed by GG, 12-Oct-2024.) (Revised by BJ, 16-Oct-2024.) |
| Ref | Expression |
|---|---|
| elabd2.ex | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| elabd2.eq | ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) |
| elabd2.is | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| elabd2 | ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elabd2.ex | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | elabd2.eq | . . . . 5 ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) | |
| 3 | 2 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})) |
| 4 | elab6g 3669 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))) | |
| 5 | 3, 4 | sylan9bb 509 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))) |
| 6 | elisset 2823 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
| 7 | elabd2.is | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
| 8 | 7 | pm5.74da 804 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 → 𝜓) ↔ (𝑥 = 𝐴 → 𝜒))) |
| 9 | 8 | albidv 1920 | . . . . . 6 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜒))) |
| 10 | 19.23v 1942 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜒) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜒)) | |
| 11 | 9, 10 | bitrdi 287 | . . . . 5 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜒))) |
| 12 | pm5.5 361 | . . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → 𝜒) ↔ 𝜒)) | |
| 13 | 11, 12 | sylan9bb 509 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒)) |
| 14 | 6, 13 | sylan2 593 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒)) |
| 15 | 5, 14 | bitrd 279 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
| 16 | 1, 15 | mpdan 687 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 |
| This theorem is referenced by: elabd3 3671 elimasng1 6105 |
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