![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rabeq2i | GIF version |
Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Ref | Expression |
---|---|
rabeq2i.1 | ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
rabeq2i | ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq2i.1 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
2 | 1 | eleq2i 2260 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
3 | rabid 2670 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | 2, 3 | bitri 184 | 1 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 {crab 2476 |
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-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-rab 2481 |
This theorem is referenced by: tfis 4611 fvmptssdm 5634 suplocsrlempr 7857 suplocsrlem 7858 |
Copyright terms: Public domain | W3C validator |