Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 2895 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | rabbida2.3 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | anbi12d 630 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
6 | 1, 5 | abbid 2884 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}) |
7 | df-rab 3144 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
8 | df-rab 3144 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)} | |
9 | 6, 7, 8 | 3eqtr4g 2878 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 {crab 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-rab 3144 |
This theorem is referenced by: smflimmpt 42961 |
Copyright terms: Public domain | W3C validator |