![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elrabd | GIF version |
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2895. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
elrabd.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
elrabd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
elrabd.3 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
elrabd | ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrabd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | elrabd.3 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝜒)) |
4 | elrabd.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
5 | 4 | elrab 2895 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
6 | 3, 5 | sylibr 134 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {crab 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2741 |
This theorem is referenced by: ctssdccl 7112 suplocexprlemru 7720 suplocexprlemloc 7722 zsupssdc 11957 uzwodc 12040 phisum 12242 odzcllem 12244 pcpremul 12295 znnen 12401 ennnfonelemj0 12404 ennnfonelemg 12406 issubmd 12870 mhmeql 12881 cdivcncfap 14126 cnopnap 14133 ivthinc 14160 limcdifap 14170 limcimolemlt 14172 dvcoapbr 14210 subctctexmid 14789 |
Copyright terms: Public domain | W3C validator |