Theorem 19.9d 2188

Description: A deduction version of one direction of 19.9 2190. (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 1778 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)

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	1	nfrd 1785	. 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
3		sp 2168	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
4	2, 3	syl6 35	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778

This theorem is referenced by:  19.9t  2189  19.9ht  2305  spimt  2377  exdistrf  2438  equvel  2447  copsexgw  5480  copsexg  5481  oprabidw  7432  19.9d2rf  32180  copsex2d  36510  wl-exeq  36893  spd  47911