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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 3695 | . 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 1540 ∈ wcel 2108 {crab 3436 | 
| 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 | 
| This theorem is referenced by: elrefrels2 38519 elrefrels3 38520 elcnvrefrels2 38535 elcnvrefrels3 38536 elsymrels2 38554 elsymrels3 38555 elsymrels4 38556 elsymrels5 38557 elrefsymrels2 38570 eltrrels2 38580 eltrrels3 38581 eleqvrels2 38593 eleqvrels3 38594 elfunsALTV 38693 eldisjs 38723 | 
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