Theorem bj-exalims

Description: Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw proves. (Contributed by BJ, 29-Sep-2019)

	Ref	Expression
Hypothesis	bj-exalims.1	⊢ ( ∃ 𝑥 𝜑 → ( ¬ 𝜒 → ∀ 𝑥 ¬ 𝜒 ) )
Assertion	bj-exalims	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-exalims.1	⊢ ( ∃ 𝑥 𝜑 → ( ¬ 𝜒 → ∀ 𝑥 ¬ 𝜒 ) )
2		bj-exalim	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) )
3		eximal	⊢ ( ( ∃ 𝑥 𝜒 → 𝜒 ) ↔ ( ¬ 𝜒 → ∀ 𝑥 ¬ 𝜒 ) )
4	1 3	sylibr	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑥 𝜒 → 𝜒 ) )
5	4	a1i	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∃ 𝑥 𝜒 → 𝜒 ) ) )
6	2 5	syldd	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) )