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Theorem equs5e 2470
 Description: A property related to substitution that unlike equs5 2472 does not require a distinctor antecedent. This proof uses ax12 2434, see equs5eALT 2374 for an alternative one using ax-12 2175 but not ax13 2382. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (New usage is discouraged.)
Assertion
Ref Expression
equs5e (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e
StepHypRef Expression
1 nfa1 2152 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 ax12 2434 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3 hbe1 2144 . . . 4 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
4319.23bi 2188 . . 3 (𝜑 → ∀𝑦𝑦𝜑)
52, 4impel 509 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
61, 5exlimi 2215 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃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  ax-13 2379 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:  sb4e  2503
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