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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 3576. (Contributed by Gino Giotto, 12-Oct-2024.) |
Ref | Expression |
---|---|
elabd2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elabd2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elabd2 | ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabd2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elab6g 3568 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))) | |
3 | 2 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))) |
4 | elisset 2812 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
5 | elabd2.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
6 | 5 | pm5.74da 804 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 → 𝜓) ↔ (𝑥 = 𝐴 → 𝜒))) |
7 | 6 | albidv 1928 | . . . . . 6 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜒))) |
8 | 19.23v 1950 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜒) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜒)) | |
9 | 7, 8 | bitrdi 290 | . . . . 5 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜒))) |
10 | pm5.5 365 | . . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → 𝜒) ↔ 𝜒)) | |
11 | 9, 10 | sylan9bb 513 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒)) |
12 | 4, 11 | sylan2 596 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒)) |
13 | 3, 12 | bitrd 282 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
14 | 1, 13 | mpdan 687 | 1 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 |
This theorem is referenced by: sbcied 3728 |
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