Theorem bj-nnfed 37634

Description: Nonfreeness implies the equivalent of ax5e 1945, 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 37633	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  Ⅎ'wnnf 37628

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-bj-nnf 37629

This theorem is used by:  bj-nnfand  37657  bj-nnford  37659  bj-nnf-spim  37676