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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 2844 | . 2 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ)) → (B = 𝐶 ↔ (𝐶 ∈ A ∧ χ))) |
4 | 3 | biimp3ar 1235 | 1 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ) ∧ (𝐶 ∈ A ∧ χ)) → B = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∃*wrmo 2303 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rmo 2308 df-v 2553 |
This theorem is referenced by: (None) |
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