Theorem bj-exalim 34813

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 1913. I propose to move to the main part: bj-exalim 34813, bj-exalimi 34814, bj-exalims 34815, bj-exalimsi 34816, bj-ax12i 34818, bj-ax12wlem 34825, bj-ax12w 34858. A new label is needed for bj-ax12i 34818 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 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Proof of Theorem bj-exalim

Step	Hyp	Ref	Expression
1		pm2.04 90	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
2	1	alimi 1814	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ∀𝑥(𝜓 → (𝜑 → 𝜒)))
3		bj-alexim 34808	. 2 ⊢ (∀𝑥(𝜓 → (𝜑 → 𝜒)) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒)))
4		pm2.04 90	. 2 ⊢ ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
5	2, 3, 4	3syl 18	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  bj-exalims  34815  bj-cbveximt  34821