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Theorem equs5eALT 2176
 Description: Alternate proof of equs5e 2347. Uses ax-12 2045 but not ax-13 2244. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equs5eALT (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5eALT
StepHypRef Expression
1 nfa1 2026 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 hbe1 2019 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
3219.23bi 2059 . . . 4 (𝜑 → ∀𝑦𝑦𝜑)
4 ax-12 2045 . . . 4 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
53, 4syl5 34 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
65imp 445 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
71, 6exlimi 2084 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1479  ∃wex 1702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708 This theorem is referenced by: (None)
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