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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 35529 | . 2 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
4 | abeqinbi.3 | . 2 ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | rabimbieq 35528 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 ∩ cin 3935 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 |
This theorem is referenced by: dfrefrels2 35768 dfcnvrefrels2 35781 dfsymrels2 35796 dftrrels2 35826 |
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