Theorem bj-nnfimd 34929

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

Hypotheses

Ref	Expression
bj-nnfimd.1	⊢ (𝜑 → Ⅎ'𝑥𝜓)
bj-nnfimd.2	⊢ (𝜑 → Ⅎ'𝑥𝜒)

Assertion

Ref	Expression
bj-nnfimd	⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))

Proof of Theorem bj-nnfimd

Step	Hyp	Ref	Expression
1		bj-nnfimd.1	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		bj-nnfimd.2	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
3		bj-nnfim 34928	. 2 ⊢ ((Ⅎ'𝑥𝜓 ∧ Ⅎ'𝑥𝜒) → Ⅎ'𝑥(𝜓 → 𝜒))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4  Ⅎ'wnnf 34905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-bj-nnf 34906

This theorem is referenced by: (None)