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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 4216 | . 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶) | 
| 3 | abeqin.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
| 4 | dfrab2 4320 | . 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶) | |
| 5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 {cab 2714 {crab 3436 ∩ cin 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 | 
| This theorem is referenced by: abeqinbi 38254 dfcnvrefrels3 38530 dffunsALTV 38684 dfdisjs 38709 | 
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