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Theorem bj-cbvalim 33145
 Description: A lemma used to prove a justification of df-bj-mo 33152 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cbvalim (∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbvalim
StepHypRef Expression
1 ax5e 2011 . . 3 (∃𝑥𝜓𝜓)
21ax-gen 1894 . 2 𝑦(∃𝑥𝜓𝜓)
3 ax-5 2009 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
4 bj-cbvalimt 33142 . . . 4 (∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → ((∀𝑥𝜑 → ∀𝑦𝑥𝜑) → (∀𝑦(∃𝑥𝜓𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
54com3l 89 . . 3 (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → ((∀𝑥𝜑 → ∀𝑦𝑥𝜑) → (∀𝑦𝑥𝜒 → (∀𝑦(∃𝑥𝜓𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
65com14 96 . 2 (∀𝑦(∃𝑥𝜓𝜓) → ((∀𝑥𝜑 → ∀𝑦𝑥𝜑) → (∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
72, 3, 6mp2 9 1 (∀𝑦𝑥𝜒 → (∀𝑦𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1654  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009 This theorem depends on definitions:  df-bi 199  df-ex 1879 This theorem is referenced by:  bj-cbvalimi  33147
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