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Theorem equs5e 2458
Description: A property related to substitution that unlike equs5 2460 does not require a distinctor antecedent. This proof uses ax12 2423, see equs5eALT 2365 for an alternative one using ax-12 2172 but not ax13 2375. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 2149 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 ax12 2423 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3 hbe1 2140 . . . 4 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
4319.23bi 2185 . . 3 (𝜑 → ∀𝑦𝑦𝜑)
52, 4impel 507 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
61, 5exlimi 2211 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172  ax-13 2372
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787
This theorem is referenced by:  sb4e  2489
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