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Theorem equs5av 2276
 Description: A property related to substitution that replaces the distinctor from equs5 2472 to a disjoint variable condition. Version of equs5a 2469 with a disjoint variable condition, which does not require ax-13 2379. See also sb56 2274. (Contributed by NM, 2-Feb-2007.) (Revised by Gino Giotto, 15-Dec-2023.)
Assertion
Ref Expression
equs5av (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs5av
StepHypRef Expression
1 nfa1 2152 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12v2 2177 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32spsd 2184 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
43imp 410 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
51, 4exlimi 2215 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  bj-equs45fv  34399
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