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Theorem equsexvw 2002
Description: Version of equsexv 2259 with a disjoint variable condition, and of equsex 2431 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2001. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
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 alinexa 1834 . . 3 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
2 equsalvw.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32notbid 319 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
43equsalvw 2001 . . 3 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ 𝜓)
51, 4bitr3i 278 . 2 (¬ ∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ 𝜓)
65con4bii 322 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  equvinv  2027  cleljust  2114  sbelx  2245  cleljustab  2799  sbhypf  3550  axsepgfromrep  5192  dfid3  5455  opeliunxp  5612  imai  5935  coi1  6108  opabex3d  7655  opabex3rd  7656  opabex3  7657  fsplit  7801  fsplitOLD  7802  mapsnend  8576  dfac5lem1  9537  dfac5lem3  9539  dffix2  33263  sscoid  33271  elfuns  33273  pmapglb  36786  polval2N  36922  opeliun2xp  44309
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