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Theorem equs5e 2337
 Description: A property related to substitution that unlike equs5 2339 does not require a distinctor antecedent. See equs5eALT 2166 for an alternate proof using ax-12 2034 but not ax13 2237. (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 2015 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 ax12 2292 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3 hbe1 2008 . . . 4 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
4319.23bi 2049 . . 3 (𝜑 → ∀𝑦𝑦𝜑)
52, 4impel 484 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
61, 5exlimi 2073 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  sb4e  2350
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