![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rabimbieq | Structured version Visualization version GIF version |
Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.) |
Ref | Expression |
---|---|
rabimbieq.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
rabimbieq.2 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabimbieq | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabimbieq.1 | . 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | rabimbieq.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | rabbiia 3418 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
4 | 1, 3 | eqtri 2819 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 {crab 3109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-9 2091 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-clab 2776 df-cleq 2788 df-rab 3114 |
This theorem is referenced by: abeqinbi 35047 dfsymrels4 35314 dfsymrels5 35315 dffunsALTV2 35448 dffunsALTV3 35449 dffunsALTV4 35450 dfdisjs2 35473 dfdisjs5 35476 |
Copyright terms: Public domain | W3C validator |