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| Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) | 
| Ref | Expression | 
|---|---|
| moi.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| moi.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| moi | ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | moi.1 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | moi.2 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
| 3 | 1, 2 | mob 3722 | . . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)) | 
| 4 | 3 | biimprd 248 | . . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝜒 → 𝐴 = 𝐵)) | 
| 5 | 4 | 3expia 1121 | . . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒 → 𝐴 = 𝐵))) | 
| 6 | 5 | impd 410 | . 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑) → ((𝜓 ∧ 𝜒) → 𝐴 = 𝐵)) | 
| 7 | 6 | 3impia 1117 | 1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 | 
| This theorem is referenced by: enqeq 10975 f1otrspeq 19466 hausflim 23990 tglineineq 28652 tglineinteq 28654 | 
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