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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 3439 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
4 | 1, 3 | eqtri 2763 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 {crab 3433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-rab 3434 |
This theorem is referenced by: dfdif3 4127 1arithufd 33556 dfrefrels3 38496 dfcnvrefrels3 38511 dfsymrels3 38528 refsymrels3 38548 dftrrels3 38558 dfeqvrels3 38571 dfdisjs3 38692 dfdisjs4 38693 isubgr0uhgr 47797 |
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