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| 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 3440 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | 
| 4 | 1, 3 | eqtri 2765 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rab 3437 | 
| This theorem is referenced by: abeqinbi 38254 dfsymrels4 38548 dfsymrels5 38549 dffunsALTV2 38685 dffunsALTV3 38686 dffunsALTV4 38687 dfdisjs2 38710 dfdisjs5 38713 | 
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