Theorem bj-exalimsi

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

	Ref	Expression
Hypotheses	bj-exalimsi.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	bj-exalimsi.2	⊢ ( ∃ 𝑥 𝜑 → ( ¬ 𝜒 → ∀ 𝑥 ¬ 𝜒 ) )
Assertion	bj-exalimsi	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-exalimsi.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		bj-exalimsi.2	⊢ ( ∃ 𝑥 𝜑 → ( ¬ 𝜒 → ∀ 𝑥 ¬ 𝜒 ) )
3	2	bj-exalims	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) )
4	3 1	mpg	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )