Theorem 19.9d 2206

Description: A deduction version of one direction of 19.9 2208. (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 1785 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 1792	. 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
3		sp 2186	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
4	2, 3	syl6 35	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785

This theorem is referenced by:  19.9t  2207  19.9ht  2321  spimt  2386  exdistrf  2447  equvel  2456  copsexgw  5425  copsexg  5426  oprabidw  7372  19.9d2rf  32440  copsex2d  37173  wl-exeq  37568  spd  49710