 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsexvw Structured version   Visualization version   GIF version

Theorem equsexvw 1919
 Description: Version of equsexv 2095 with a dv condition, which requires fewer axioms. See also equsex 2281. (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 667 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1764 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 1877 . . 3 𝑥 𝑥 = 𝑦
5 19.41v 1901 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
64, 5mpbiran 955 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
73, 6bitri 263 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  cleljust  1985  sbhypf  3226  axsep  4708  dfid3  4954  opeliunxp  5093  imai  5397  coi1  5568  bj-axsep  31981
 Copyright terms: Public domain W3C validator