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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equs45fv | Structured version Visualization version GIF version |
Description: Version of equs45f 2397 with a disjoint variable condition, which does not require ax-13 2302. Note that the version of equs5 2398 with a disjoint variable condition is actually sb56 2207 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-equs45fv.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
bj-equs45fv | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equs45fv.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nf5ri 2124 | . . . . 5 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | anim2i 608 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
4 | 3 | eximi 1798 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
5 | equs5a 2395 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
7 | equs4v 1960 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
8 | 6, 7 | impbii 201 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1506 ∃wex 1743 Ⅎwnf 1747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-10 2080 ax-12 2107 ax-13 2302 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ex 1744 df-nf 1748 |
This theorem is referenced by: (None) |
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