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Theorem equsexv 2255
Description: An equivalence related to implicit substitution. Version of equsex 2413 with a disjoint variable condition, which does not require ax-13 2367. See equsexvw 2004 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2254. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2132. (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 476 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimi 2205 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
51, 2equsalv 2254 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
6 equs4v 1999 . . 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 395  wal 1535  wex 1777  wnf 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1778  df-nf 1782
This theorem is referenced by:  sb5OLD  2264  equsexhv  2284  cleljustALT2  2358  sb10f  2527  dprd2d2  19675  poimirlem25  35830
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