Theorem bj-nnfim1

Description: A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023)

	Ref	Expression
Assertion	bj-nnfim1	⊢ ( ( Ⅎ' 𝑥 𝜑 ∧ Ⅎ' 𝑥 𝜓 ) → ( ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
2		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜓 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
3		imim12	⊢ ( ( ∃ 𝑥 𝜑 → 𝜑 ) → ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) )
4	3	imp	⊢ ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )
5	1 2 4	syl2an	⊢ ( ( Ⅎ' 𝑥 𝜑 ∧ Ⅎ' 𝑥 𝜓 ) → ( ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )