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Theorem equsexvw 1930
Description: Version of equsexv 2107 with a dv condition, and of equsex 2290 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 1929. (Contributed by BJ, 31-May-2019.)
Hypothesis
Ref Expression
equsalvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexvw (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem equsexvw
StepHypRef Expression
1 equsalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.32i 668 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1772 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 1888 . . 3 𝑥 𝑥 = 𝑦
5 19.41v 1912 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
64, 5mpbiran 952 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
73, 6bitri 264 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  cleljust  1996  sbhypf  3248  axsep  4771  dfid3  5015  opeliunxp  5160  imai  5466  coi1  5639  bj-axsep  32768
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