Theorem 19.9d 2245

Description: A deduction version of one direction of 19.9 2247. (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 1811 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 1818	. 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
3		sp 2225	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
4	2, 3	syl6 36	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  19.9t  2246  19.9ht  2359  spimt  2424  exdistrf  2485  equvel  2494  copsexgwOLD  5474  copsexg  5475  oprabidw  7442  19.9d2rf  32756  copsex2d  37670  wl-exeq  38076  spd  50340