Theorem bj-aleximiALT 33996

Description: Alternate proof of aleximi 1831 from exim 1833, which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-aleximiALT.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-aleximiALT	⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem bj-aleximiALT

Step	Hyp	Ref	Expression
1		bj-aleximiALT.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alimi 1811	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))
3		bj-eximALT 33995	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒))
4	2, 3	syl 17	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by: (None)