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Theorem bj-cbvexim 34093
Description: A lemma used to prove bj-cbvex 34097 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cbvexim (∀𝑥𝑦𝜒 → (∀𝑥𝑦(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-cbvexim
StepHypRef Expression
1 ax5e 1913 . 2 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
2 ax-5 1911 . . 3 (𝜑 → ∀𝑦𝜑)
32ax-gen 1797 . 2 𝑥(𝜑 → ∀𝑦𝜑)
4 bj-cbveximt 34087 . . . 4 (∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
54com3l 89 . . 3 (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝑦𝜒 → ((∃𝑥𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
65com14 96 . 2 ((∃𝑥𝑦𝜓 → ∃𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
71, 3, 6mp2 9 1 (∀𝑥𝑦𝜒 → (∀𝑥𝑦(𝜒 → (𝜑𝜓)) → (∃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-5 1911
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  bj-cbveximi  34095
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