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Metamath Proof Explorer

Theorem bj-exalimi

Description: An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim (using mpg ) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw proves. (Contributed by BJ, 29-Sep-2019)

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

Proof

Step	Hyp	Ref	Expression
1		bj-exalimi.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	com12	⊢ ( 𝜓 → ( 𝜑 → 𝜒 ) )
3	2	aleximi	⊢ ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜒 ) )
4	3	com12	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )