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Theorem equs5e 2377
 Description: A property related to substitution that unlike equs5 2379 does not require a distinctor antecedent. See equs5eALT 2214 for an alternate proof using ax-12 2087 but not ax13 2285. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
Assertion
Ref Expression
equs5e (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e
StepHypRef Expression
1 nfa1 2068 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 ax12 2340 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3 hbe1 2061 . . . 4 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
4319.23bi 2099 . . 3 (𝜑 → ∀𝑦𝑦𝜑)
52, 4impel 484 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
61, 5exlimi 2124 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by:  sb4e  2390
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