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Theorem equsexv 2269
Description: An equivalence related to implicit substitution. Version of equsex 2419 with a disjoint variable condition, which does not require ax-13 2373. See equsexvw 2016 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2268. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
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 sbalex 2244 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 equsalv.nf . . 3 𝑥𝜓
3 equsalv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsalv 2268 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
51, 4bitri 278 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540  wex 1786  wnf 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791
This theorem is referenced by:  sb5OLD  2277  equsexhv  2297  cleljustALT2  2366  sb10f  2533  dprd2d2  19297  poimirlem25  35457
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