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Theorem bj-cbvexiw 33901
Description: Change bound variable. This is to cbvexvw 2035 what cbvaliw 2004 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.)
Hypotheses
Ref Expression
bj-cbvexiw.1 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
bj-cbvexiw.2 (𝜑 → ∀𝑦𝜑)
bj-cbvexiw.3 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbvexiw (∃𝑥𝜑 → ∃𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-cbvexiw
StepHypRef Expression
1 bj-cbvexiw.1 . 2 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
2 bj-cbvexiw.2 . . 3 (𝜑 → ∀𝑦𝜑)
3 bj-cbvexiw.3 . . 3 (𝑦 = 𝑥 → (𝜑𝜓))
42, 3spimew 1965 . 2 (𝜑 → ∃𝑦𝜓)
51, 4bj-sylge 33854 1 (∃𝑥𝜑 → ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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-6 1961
This theorem depends on definitions:  df-bi 208  df-ex 1772
This theorem is referenced by:  bj-cbvexivw  33902  bj-cbvexw  33906
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