![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elab3 | GIF version |
Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
Ref | Expression |
---|---|
elab3.1 | ⊢ (𝜓 → 𝐴 ∈ V) |
elab3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elab3 | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab3.1 | . 2 ⊢ (𝜓 → 𝐴 ∈ V) | |
2 | elab3.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | elab3g 2903 | . 2 ⊢ ((𝜓 → 𝐴 ∈ V) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {cab 2175 Vcvv 2752 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: elrnmpo 6011 isfi 6788 addnqprlemfl 7589 addnqprlemfu 7590 mulnqprlemfl 7605 mulnqprlemfu 7606 4sqlem2 12424 istps 14009 |
Copyright terms: Public domain | W3C validator |