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Theorem equsex 2417
Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. See equsexvw 2008 and equsexv 2259 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2416. See equsexALT 2418 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsal.1 𝑥𝜓
equsal.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsex (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsex
StepHypRef Expression
1 equsal.1 . . 3 𝑥𝜓
2 equsal.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpa 477 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimi 2210 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
51, 2equsal 2416 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
6 equs4 2415 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6sylbir 234 . 2 (𝜓 → ∃𝑥(𝑥 = 𝑦𝜑))
84, 7impbii 208 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786
This theorem is referenced by:  equsexh  2420  sb5rf  2466
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