![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > moi2 | GIF version |
Description: Consequence of "at most one". (Contributed by NM, 29-Jun-2008.) |
Ref | Expression |
---|---|
moi2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
moi2 | ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moi2.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | mob2 2932 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) |
3 | 2 | 3expa 1205 | . . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) |
4 | 3 | biimprd 158 | . 2 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓 → 𝑥 = 𝐴)) |
5 | 4 | impr 379 | 1 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃*wmo 2039 ∈ wcel 2160 |
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-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: fsum3 11413 fprodseq 11609 txcn 14172 |
Copyright terms: Public domain | W3C validator |