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Theorem bj-exalim 34022
Description: Distribute quantifiers over a nested implication.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1912. I propose to move to the main part: bj-exalim 34022, bj-exalimi 34023, bj-exalims 34024, bj-exalimsi 34025, bj-ax12i 34027, bj-ax12wlem 34034, bj-ax12w 34067, and remove equs3OLD 1966. A new label is needed for bj-ax12i 34027 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1967 and spimfw 1969 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.)

Assertion
Ref Expression
bj-exalim (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Proof of Theorem bj-exalim
StepHypRef Expression
1 pm2.04 90 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
21alimi 1813 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → ∀𝑥(𝜓 → (𝜑𝜒)))
3 bj-alexim 34017 . 2 (∀𝑥(𝜓 → (𝜑𝜒)) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒)))
4 pm2.04 90 . 2 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
52, 3, 43syl 18 1 (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
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
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  bj-exalims  34024  bj-cbveximt  34030
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