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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabbida2 | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
rabbida2.1 | ⊢ Ⅎ𝑥𝜑 |
rabbida2.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
rabbida2.3 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabbida2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbida2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rabbida2.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2822 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | rabbida2.3 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | anbi12d 631 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
6 | 1, 5 | abbid 2807 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}) |
7 | df-rab 3404 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
8 | df-rab 3404 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)} | |
9 | 6, 7, 8 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 {cab 2713 {crab 3403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-12 2170 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 |
This theorem is referenced by: smflimmpt 44685 |
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