Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqrrabd | Structured version Visualization version GIF version |
Description: Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
Ref | Expression |
---|---|
eqrrabd.1 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
eqrrabd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) |
Ref | Expression |
---|---|
eqrrabd | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1922 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfrab1 3286 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓} | |
4 | eqrrabd.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
5 | 4 | sseld 3886 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
6 | 5 | pm4.71rd 566 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
7 | eqrrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) | |
8 | 7 | pm5.32da 582 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
9 | 6, 8 | bitrd 282 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | rabid 3280 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) | |
11 | 9, 10 | bitr4di 292 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) |
12 | 1, 2, 3, 11 | eqrd 3906 | 1 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3055 ⊆ wss 3853 |
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-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-rab 3060 df-v 3400 df-in 3860 df-ss 3870 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |