Theorem bj-nnfed 34914

Description: Nonfreeness implies the equivalent of ax5e 1915, deduction form. (Contributed by BJ, 2-Dec-2023.)

Hypothesis

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

Assertion

Ref	Expression
bj-nnfed	⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))

Proof of Theorem bj-nnfed

Step	Hyp	Ref	Expression
1		bj-nnfed.1	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		bj-nnfe 34913	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))

Syntax hints:   → wi 4  ∃wex 1782  Ⅎ'wnnf 34905

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 397  df-bj-nnf 34906

This theorem is referenced by:  bj-nnfand  34931  bj-nnford  34933