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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eu3f | Structured version Visualization version GIF version |
Description: Version of eu3v 2588 where the disjoint variable condition is replaced with a non-freeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2588. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
bj-eu3f.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
bj-eu3f | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2587 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
2 | bj-eu3f.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | mof 2578 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | 3 | anbi2i 616 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
5 | 1, 4 | bitri 267 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1599 ∃wex 1823 Ⅎwnf 1827 ∃*wmo 2549 ∃!weu 2586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-11 2150 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-mo 2551 df-eu 2587 |
This theorem is referenced by: (None) |
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