Theorem 19.9d 2058

Description: A deduction version of one direction of 19.9 2060. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Hypothesis

Ref	Expression
19.9d.1	⊢ (𝜓 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
19.9d	⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Proof of Theorem 19.9d

Step	Hyp	Ref	Expression
1		19.9d.1	. . 3 ⊢ (𝜓 → Ⅎ𝑥𝜑)
2		df-nf 1701	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	sylib 207	. 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		sp 2041	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
5	3, 4	syl6 34	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  19.9t  2059  exdistrf  2321  equvel  2335  copsexg  4882  19.9d2rf  28702  wl-exeq  32500  spd  42223