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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3415 | . 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 205 = wceq 1539 {crab 3303 |
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 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-rab 3306 |
This theorem is referenced by: dfrefrels3 36728 dfcnvrefrels3 36743 dfsymrels3 36760 refsymrels3 36780 dftrrels3 36790 dfeqvrels3 36803 dfdisjs3 36924 dfdisjs4 36925 |
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