Theorem 19.9dOLDOLD 2245

Description: Obsolete version of 19.9d 2244 as of 8-Jul-2022. (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 1883 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.9d.1	⊢ (𝜓 → Ⅎ𝑥𝜑)

Assertion

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

Proof of Theorem 19.9dOLDOLD

Step	Hyp	Ref	Expression
1		19.9d.1	. . 3 ⊢ (𝜓 → Ⅎ𝑥𝜑)
2		df-nf 1883	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	sylib 210	. 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		sp 2224	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
5	3, 4	syl6 35	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1654  ∃wex 1878  Ⅎwnf 1882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-ex 1879  df-nf 1883

This theorem is referenced by: (None)