Theorem bj-nnfimd 37435

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 37434	. 2 ⊢ ((Ⅎ'𝑥𝜓 ∧ Ⅎ'𝑥𝜒) → Ⅎ'𝑥(𝜓 → 𝜒))
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  Ⅎ'wnnf 37408

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37409

This theorem is used by: (None)