Theorem bj-nnfim1 34853

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

Assertion

Ref	Expression
bj-nnfim1	⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-nnfim1

Step	Hyp	Ref	Expression
1		bj-nnfe 34840	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
2		bj-nnfa 34837	. 2 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3		imim12 105	. . 3 ⊢ ((∃𝑥𝜑 → 𝜑) → ((𝜓 → ∀𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))))
4	3	imp 406	. 2 ⊢ (((∃𝑥𝜑 → 𝜑) ∧ (𝜓 → ∀𝑥𝜓)) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5	1, 2, 4	syl2an 595	1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783  Ⅎ'wnnf 34832

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-bj-nnf 34833

This theorem is referenced by:  bj-nnfim  34855