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Theorem equs5e 2438
 Description: A property related to substitution that unlike equs5 2440 does not require a distinctor antecedent. See equs5eALT 2342 for an alternate proof using ax-12 2141 but not ax13 2347. (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 2121 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 ax12 2402 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3 hbe1 2114 . . . 4 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
4319.23bi 2154 . . 3 (𝜑 → ∀𝑦𝑦𝜑)
52, 4impel 506 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
61, 5exlimi 2182 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520  ∃wex 1761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766 This theorem is referenced by:  sb4e  2478
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