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Theorem equsexv 2147
 Description: Version of equsex 2328 with a dv condition, which does not require ax-13 2282. See equsexvw 1978 for a version with two dv conditions requiring fewer axioms. See also the dual form equsalv 2146. (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 670 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1814 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 1947 . . 3 𝑥 𝑥 = 𝑦
5 equsalv.nf . . . 4 𝑥𝜓
6519.41 2141 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 973 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 264 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∃wex 1744  Ⅎwnf 1748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by:  equsexhv  2162  sb56  2188  cleljustALT2  2222  sb10f  2484  dprd2d2  18489  poimirlem25  33564
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