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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 3631 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓)) |
4 | 3 | biancomi 466 | 1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 |
This theorem is referenced by: elrefrels2 35917 elrefrels3 35918 elcnvrefrels2 35930 elcnvrefrels3 35931 elsymrels2 35949 elsymrels3 35950 elsymrels4 35951 elsymrels5 35952 elrefsymrels2 35965 eltrrels2 35975 eltrrels3 35976 eleqvrels2 35987 eleqvrels3 35988 elfunsALTV 36085 eldisjs 36115 |
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