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Theorem equsexv 2260
Description: An equivalence related to implicit substitution. Version of equsex 2417 with a disjoint variable condition, which does not require ax-13 2371. See equsexvw 2009 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2259. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2138. (Revised by Gino Giotto, 18-Nov-2024.)
Hypotheses
Ref Expression
equsalv.nf 𝑥𝜓
equsalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexv (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexv
StepHypRef Expression
1 equsalv.nf . . 3 𝑥𝜓
2 equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpa 478 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimi 2211 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
51, 2equsalv 2259 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
6 equs4v 2004 . . 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 397  wal 1540  wex 1782  wnf 1786
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-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787
This theorem is referenced by:  sb5OLD  2269  equsexhv  2289  cleljustALT2  2362  sb10f  2531  dprd2d2  19830  poimirlem25  36132
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