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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 4203 | . 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶) |
3 | abeqin.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
4 | dfrab2 4305 | . 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4i 2764 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2703 {crab 3426 ∩ cin 3942 |
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-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 |
This theorem is referenced by: abeqinbi 37632 dfcnvrefrels3 37910 dffunsALTV 38064 dfdisjs 38089 |
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