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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 3876 | . 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) |
8 | 1, 7 | mpbird 259 | 1 ⊢ (𝜑 → 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃*wrmo 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2800 df-cleq 2814 df-clel 2893 df-rmo 3146 df-v 3496 |
This theorem is referenced by: lmieu 26570 |
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