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Mirrors > Home > MPE Home > Th. List > rmob2 | 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) |
Ref | Expression |
---|---|
rmob2 | ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoi2.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
2 | rmoi2.3 | . . . 4 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) | |
3 | df-rmo 2949 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
4 | 2, 3 | sylib 208 | . . 3 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
5 | rmoi2.4 | . . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
6 | rmoi2.5 | . . 3 ⊢ (𝜑 → 𝜓) | |
7 | eleq1 2718 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
8 | rmoi2.1 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) | |
9 | 7, 8 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒))) |
10 | 9 | mob2 3419 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒))) |
11 | 1, 4, 5, 6, 10 | syl112anc 1370 | . 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ (𝐵 ∈ 𝐴 ∧ 𝜒))) |
12 | 1, 11 | mpbirand 529 | 1 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∃*wmo 2499 ∃*wrmo 2944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rmo 2949 df-v 3233 |
This theorem is referenced by: rmoi2 3565 |
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