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Mirrors > Home > MPE Home > Th. List > rmoi2 | Structured version Visualization version GIF version |
Description: Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019.) |
Ref | Expression |
---|---|
rmoi2.1 | ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) |
rmoi2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
rmoi2.3 | ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) |
rmoi2.4 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) |
rmoi2.5 | ⊢ (𝜑 → 𝜓) |
rmoi2.6 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
rmoi2 | ⊢ (𝜑 → 𝑥 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoi2.6 | . 2 ⊢ (𝜑 → 𝜒) | |
2 | rmoi2.1 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | rmoi2.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
4 | rmoi2.3 | . . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) | |
5 | rmoi2.4 | . . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
6 | rmoi2.5 | . . 3 ⊢ (𝜑 → 𝜓) | |
7 | 2, 3, 4, 5, 6 | rmob2 3901 | . 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) |
8 | 1, 7 | mpbird 257 | 1 ⊢ (𝜑 → 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∃*wrmo 3377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-rmo 3378 df-v 3480 |
This theorem is referenced by: lmieu 28807 |
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