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Theorem equsexv 2291
Description: Version of equsex 2425 with a disjoint variable condition, which does not require ax-13 2377. See equsexvw 2104 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2290. (Contributed 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 equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.32i 571 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1944 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 2074 . . 3 𝑥 𝑥 = 𝑦
5 equsalv.nf . . . 4 𝑥𝜓
6519.41 2270 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 701 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 267 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wex 1875  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880
This theorem is referenced by:  sb56  2297  equsexhv  2316  cleljustALT2  2373  sb10f  2576  dprd2d2  18759  poimirlem25  33923
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