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Mirrors > Home > MPE Home > Th. List > moi2 | Structured version Visualization version 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 3704 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) |
3 | 2 | 3expa 1115 | . . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) |
4 | 3 | biimprd 247 | . 2 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓 → 𝑥 = 𝐴)) |
5 | 4 | impr 454 | 1 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃*wmo 2524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 |
This theorem is referenced by: fsum 15664 fprod 15883 txcn 23454 haustsms2 23965 |
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