Theorem bj-nnfim 36769

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 1877	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		bj-nnfim2 36768	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓)))
3	1, 2	biimtrid 242	. 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)))
4		bj-nnfim1 36767	. . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5		19.38 1839	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
6	4, 5	syl6 35	. 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
7		df-bj-nnf 36747	. 2 ⊢ (Ⅎ'𝑥(𝜑 → 𝜓) ↔ ((∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))))
8	3, 6, 7	sylanbrc 583	1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎ'wnnf 36746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-bj-nnf 36747

This theorem is referenced by:  bj-nnfimd  36770  bj-nnfbit  36775  bj-nnfbid  36776