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| Description: A property related to substitution that replaces the distinctor from equs5 2464 to a disjoint variable condition. Version of equs5a 2461 with a disjoint variable condition, which does not require ax-13 2376. See also sbalex 2241. (Contributed by NM, 2-Feb-2007.) (Revised by GG, 15-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| equs5av | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfa1 2150 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
| 2 | ax12v2 2178 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
| 3 | 2 | spsd 2186 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | 
| 4 | 3 | imp 406 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| 5 | 1, 4 | exlimi 2216 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: bj-equs45fv 36813 | 
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