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Theorem bj-cbvexiw 34268
 Description: Change bound variable. This is to cbvexvw 2044 what cbvaliw 2013 is to cbvalvw 2043. TODO: move after cbvalivw 2014. (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 1974 . 2 (𝜑 → ∃𝑦𝜓)
51, 4bj-sylge 34221 1 (∃𝑥𝜑 → ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-6 1970 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  bj-cbvexivw  34269  bj-cbvexw  34273
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