Theorem bj-nnfead 34844

Description: Nonfreeness implies the equivalent of ax5ea 1917, deduction form. (Contributed by BJ, 2-Dec-2023.)

Hypothesis

Ref	Expression
bj-nnfead.1	⊢ (𝜑 → Ⅎ'𝑥𝜓)

Assertion

Ref	Expression
bj-nnfead	⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Proof of Theorem bj-nnfead

Step	Hyp	Ref	Expression
1		bj-nnfead.1	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		bj-nnfea 34843	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀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: (None)