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Mirrors > Home > MPE Home > Th. List > Mathboxes > abeqinbi | Structured version Visualization version GIF version |
Description: Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.) |
Ref | Expression |
---|---|
abeqinbi.1 | ⊢ 𝐴 = (𝐵 ∩ 𝐶) |
abeqinbi.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
abeqinbi.3 | ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
abeqinbi | ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqinbi.1 | . . 3 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
2 | abeqinbi.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
3 | 1, 2 | abeqin 36525 | . 2 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
4 | abeqinbi.3 | . 2 ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | rabimbieq 36524 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 {cab 2713 {crab 3403 ∩ cin 3897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 df-v 3443 df-in 3905 |
This theorem is referenced by: dfrefrels2 36788 dfcnvrefrels2 36803 dfsymrels2 36820 dftrrels2 36850 |
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