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| Mirrors > Home > MPE Home > Th. List > rmoi | Structured version Visualization version GIF version | ||
| Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| rmoi.b | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | 
| rmoi.c | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rmoi | ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rmoi.b | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 2 | rmoi.c | . . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) | |
| 3 | 1, 2 | rmob 3889 | . 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) | 
| 4 | 3 | biimp3ar 1471 | 1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∃*wrmo 3378 | 
| 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-rmo 3379 df-v 3481 | 
| This theorem is referenced by: eqsqrtd 15407 efgred2 19772 0frgp 19798 frgpnabllem2 19893 frgpcyg 21593 cdleme0moN 40228 proot1mul 43211 | 
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