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Theorem bj-cbvexw 31685
Description: Change bound variable. This is to cbvexvw 1919 what cbvalw 1917 is to cbvalvw 1918. (Contributed by BJ, 17-Mar-2020.)
Hypotheses
Ref Expression
bj-cbvexw.1 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
bj-cbvexw.2 (𝜑 → ∀𝑦𝜑)
bj-cbvexw.3 (∃𝑦𝑥𝜑 → ∃𝑥𝜑)
bj-cbvexw.4 (𝜓 → ∀𝑥𝜓)
bj-cbvexw.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbvexw (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-cbvexw
StepHypRef Expression
1 bj-cbvexw.1 . . 3 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
2 bj-cbvexw.2 . . 3 (𝜑 → ∀𝑦𝜑)
3 bj-cbvexw.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
43equcoms 1897 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
54biimpd 217 . . 3 (𝑦 = 𝑥 → (𝜑𝜓))
61, 2, 5bj-cbvexiw 31680 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
7 bj-cbvexw.3 . . 3 (∃𝑦𝑥𝜑 → ∃𝑥𝜑)
8 bj-cbvexw.4 . . 3 (𝜓 → ∀𝑥𝜓)
93biimprd 236 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
107, 8, 9bj-cbvexiw 31680 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
116, 10impbii 197 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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