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Mirrors > Home > NFE Home > Th. List > moi | GIF version |
Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Ref | Expression |
---|---|
moi.1 | ⊢ (x = A → (φ ↔ ψ)) |
moi.2 | ⊢ (x = B → (φ ↔ χ)) |
Ref | Expression |
---|---|
moi | ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ (ψ ∧ χ)) → A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moi.1 | . . . . . 6 ⊢ (x = A → (φ ↔ ψ)) | |
2 | moi.2 | . . . . . 6 ⊢ (x = B → (φ ↔ χ)) | |
3 | 1, 2 | mob 3018 | . . . . 5 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ)) |
4 | 3 | biimprd 214 | . . . 4 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (χ → A = B)) |
5 | 4 | 3expia 1153 | . . 3 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ) → (ψ → (χ → A = B))) |
6 | 5 | imp3a 420 | . 2 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ) → ((ψ ∧ χ) → A = B)) |
7 | 6 | 3impia 1148 | 1 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ (ψ ∧ χ)) → A = B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ∃*wmo 2205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 |
This theorem is referenced by: (None) |
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