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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 3682 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓)) |
4 | 3 | biancomi 465 | 1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 |
This theorem is referenced by: elrefrels2 35751 elrefrels3 35752 elcnvrefrels2 35764 elcnvrefrels3 35765 elsymrels2 35783 elsymrels3 35784 elsymrels4 35785 elsymrels5 35786 elrefsymrels2 35799 eltrrels2 35809 eltrrels3 35810 eleqvrels2 35821 eleqvrels3 35822 elfunsALTV 35919 eldisjs 35949 |
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