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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrab2w | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction. This is to elrab2 3687 what elab2gw 41409 is to elab2g 3671. (Contributed by SN, 2-Sep-2024.) |
Ref | Expression |
---|---|
elrab2w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
elrab2w.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
elrab2w.3 | ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
elrab2w | ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3493 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 482 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → 𝐴 ∈ V) |
4 | eleq1w 2817 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
5 | elrab2w.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓))) |
7 | eleq1 2822 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
8 | elrab2w.2 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
9 | 7, 8 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))) |
10 | elrab2w.3 | . . . 4 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
11 | df-rab 3434 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
12 | 10, 11 | eqtri 2761 | . . 3 ⊢ 𝐶 = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} |
13 | 6, 9, 12 | elab2gw 41409 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))) |
14 | 1, 3, 13 | pm5.21nii 380 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 {crab 3433 Vcvv 3475 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 |
This theorem is referenced by: (None) |
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