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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 3497 | . 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) |
8 | 1, 7 | mpbird 246 | 1 ⊢ (𝜑 → 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃*wrmo 2899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rmo 2904 df-v 3175 |
This theorem is referenced by: lmieu 25476 |
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