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Mirrors > Home > ILE Home > Th. List > rmoi | 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 2967 | . 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) |
4 | 3 | biimp3ar 1305 | 1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 943 = wceq 1312 ∈ wcel 1461 ∃*wrmo 2391 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rmo 2396 df-v 2657 |
This theorem is referenced by: (None) |
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