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| Mirrors > Home > MPE Home > Th. List > rabbieq | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| rabbieq.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | 
| rabbieq.2 | ⊢ (𝜑 ↔ 𝜓) | 
| Ref | Expression | 
|---|---|
| rabbieq | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabbieq.1 | . 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
| 2 | rabbieq.2 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
| 3 | 2 | rabbii 3441 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | 
| 4 | 1, 3 | eqtri 2764 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 {crab 3435 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-rab 3436 | 
| This theorem is referenced by: dfdif3 4116 1arithufd 33577 dfrefrels3 38516 dfcnvrefrels3 38531 dfsymrels3 38548 refsymrels3 38568 dftrrels3 38578 dfeqvrels3 38591 dfdisjs3 38712 dfdisjs4 38713 isubgr0uhgr 47864 | 
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