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Mirrors > Home > MPE Home > Th. List > equs5av | Structured version Visualization version GIF version |
Description: A property related to substitution that replaces the distinctor from equs5 2463 to a disjoint variable condition. Version of equs5a 2460 with a disjoint variable condition, which does not require ax-13 2375. See also sbalex 2240. (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 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
2 | ax12v2 2177 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 2 | spsd 2185 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | 3 | imp 406 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 1, 4 | exlimi 2215 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: bj-equs45fv 36794 |
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