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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 3471 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
4 | 1, 3 | eqtri 2841 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 {crab 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-rab 3144 |
This theorem is referenced by: dfrefrels3 35634 dfcnvrefrels3 35647 dfsymrels3 35662 refsymrels3 35682 dftrrels3 35692 dfeqvrels3 35704 dfdisjs3 35823 dfdisjs4 35824 |
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