Theorem bj-nnfim 37634

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

Assertion

Ref	Expression
bj-nnfim	⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))

Proof of Theorem bj-nnfim

Step	Hyp	Ref	Expression
1		19.35 1910	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		bj-nnfim2 37624	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓)))
3	1, 2	biimtrid 245	. 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)))
4		bj-nnfim1 37623	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5		19.38 1872	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
6	4, 5	syl6 36	. 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
7		df-bj-nnf 37609	. 2 ⊢ (Ⅎ'𝑥(𝜑 → 𝜓) ↔ ((∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))))
8	3, 6, 7	sylanbrc 595	1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37608

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 37609

This theorem is used by:  bj-nnfimd  37635  bj-nnfbit  37640  bj-nnfbid  37641