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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 2929 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) |
3 | 2 | 3expa 1204 | . . 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 1363 ∃*wmo 2037 ∈ wcel 2158 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 |
This theorem is referenced by: fsum3 11409 fprodseq 11605 txcn 14128 |
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