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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabeqel | Structured version Visualization version GIF version |
Description: Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.) |
Ref | Expression |
---|---|
rabeqel.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
rabeqel.2 | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabeqel | ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqel.2 | . . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓)) | |
2 | rabeqel.1 | . . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
3 | 1, 2 | elrab2 3698 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓)) |
4 | 3 | biancomi 462 | 1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 |
This theorem is referenced by: elrefrels2 38500 elrefrels3 38501 elcnvrefrels2 38516 elcnvrefrels3 38517 elsymrels2 38535 elsymrels3 38536 elsymrels4 38537 elsymrels5 38538 elrefsymrels2 38551 eltrrels2 38561 eltrrels3 38562 eleqvrels2 38574 eleqvrels3 38575 elfunsALTV 38674 eldisjs 38704 |
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