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Mathbox for Peter Mazsa |
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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 3422 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
4 | 1, 3 | eqtri 2753 | 1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {crab 3418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-rab 3419 |
This theorem is referenced by: abeqinbi 37857 dfsymrels4 38151 dfsymrels5 38152 dffunsALTV2 38288 dffunsALTV3 38289 dffunsALTV4 38290 dfdisjs2 38313 dfdisjs5 38316 |
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