Theorem bj-alexim 34836

Description: Closed form of aleximi 1830. Note: this proof is shorter, so aleximi 1830 could be deduced from it (exim 1832 would have to be proved first, see bj-eximALT 34850 but its proof is shorter (currently almost a subproof of aleximi 1830)). (Contributed by BJ, 8-Nov-2021.)

Assertion

Ref	Expression
bj-alexim	⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))

Proof of Theorem bj-alexim

Step	Hyp	Ref	Expression
1		alim 1808	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)))
2		exim 1832	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒))
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 206  df-ex 1778

This theorem is referenced by:  bj-exalim  34841  bj-cbveximt  34849