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Mirrors > Home > MPE Home > Th. List > rabeq0w | Structured version Visualization version GIF version |
Description: Condition for a restricted class abstraction to be empty. Version of rabeq0 4285 using implicit substitution, which does not require ax-10 2143, ax-11 2160, ax-12 2177, but requires ax-8 2114. (Contributed by Gino Giotto, 30-Sep-2024.) |
Ref | Expression |
---|---|
rabeq0w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabeq0w | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2813 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | rabeq0w.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
4 | 3 | ab0w 4274 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = ∅ ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ 𝜓)) |
5 | df-rab 3060 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 5 | eqeq1i 2741 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = ∅) |
7 | raln 3068 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐴 ∧ 𝜓)) | |
8 | 4, 6, 7 | 3bitr4i 306 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 {cab 2714 ∀wral 3051 {crab 3055 ∅c0 4223 |
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-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-dif 3856 df-nul 4224 |
This theorem is referenced by: dffr2 5500 frc 5502 frirr 5513 |
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