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Metamath Proof Explorer

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	$⊢ φ \to Ⅎ' x ψ$
	bj-nnfimd.2	$⊢ φ \to Ⅎ' x χ$
Assertion	bj-nnfimd	$⊢ φ \to Ⅎ' x (ψ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfimd.1	$⊢ φ \to Ⅎ' x ψ$
2		bj-nnfimd.2	$⊢ φ \to Ⅎ' x χ$
3		bj-nnfim	$⊢ (Ⅎ' x ψ \land Ⅎ' x χ) \to Ⅎ' x (ψ \to χ)$
4	1 2 3	syl2anc	$⊢ φ \to Ⅎ' x (ψ \to χ)$