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Metamath Proof Explorer

Theorem bj-alexim

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

		Ref	Expression
	Assertion	bj-alexim	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		alim	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) )
2		exim	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) )