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Theorem equsexvw 2090
 Description: Version of equsexv 2265 with a dv condition, and of equsex 2447 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 2089. (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 564 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1924 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 2059 . . 3 𝑥 𝑥 = 𝑦
5 19.41v 2029 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
64, 5mpbiran 688 . 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 382  ∃wex 1852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by:  equvinv  2115  cleljust  2153  sbhypf  3405  axsep  4914  dfid3  5158  opeliunxp  5310  imai  5619  coi1  5795  elfuns  32359  bj-axsep  33129
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