Theorem bj-nnfimd

Description: Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Hypotheses	bj-nnfimd.1	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜓 )
	bj-nnfimd.2	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
Assertion	bj-nnfimd	⊢ ( 𝜑 → Ⅎ' 𝑥 ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfimd.1	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜓 )
2		bj-nnfimd.2	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
3		bj-nnfim	⊢ ( ( Ⅎ' 𝑥 𝜓 ∧ Ⅎ' 𝑥 𝜒 ) → Ⅎ' 𝑥 ( 𝜓 → 𝜒 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → Ⅎ' 𝑥 ( 𝜓 → 𝜒 ) )