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Theorem cbvexvw 1930
Description: Change bound variable. See cbvexv 1928 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1458. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvalvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexvw (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvexvw
StepHypRef Expression
1 cbvalvw.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
21biimpd 144 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
32equcoms 1718 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
43spimev 1871 . . 3 (𝜑 → ∃𝑦𝜓)
54exlimiv 1608 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
61biimprd 158 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76spimev 1871 . . 3 (𝜓 → ∃𝑥𝜑)
87exlimiv 1608 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
95, 8impbii 126 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471
This theorem is referenced by:  cbvrexvw  2720  prodmodc  11599
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