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Mirrors > Home > MPE Home > Th. List > Mathboxes > abeqin | Structured version Visualization version GIF version |
Description: Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.) |
Ref | Expression |
---|---|
abeqin.1 | ⊢ 𝐴 = (𝐵 ∩ 𝐶) |
abeqin.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
abeqin | ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqin.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
2 | 1 | ineq1i 4187 | . 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶) |
3 | abeqin.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
4 | dfrab2 4281 | . 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4i 2856 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2801 {crab 3144 ∩ cin 3937 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-in 3945 |
This theorem is referenced by: abeqinbi 35517 dfcnvrefrels3 35769 dffunsALTV 35918 dfdisjs 35943 |
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