Theorem 19.9dOLD 2191

Description: Obsolete proof of 19.9d 2058 as of 6-Oct-2021. (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.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.9dOLD.1	⊢ (𝜓 → Ⅎ𝑥𝜑)

Assertion

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

Proof of Theorem 19.9dOLD

Step	Hyp	Ref	Expression
1		19.9dOLD.1	. . 3 ⊢ (𝜓 → Ⅎ𝑥𝜑)
2		df-nfOLD 1712	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3	1, 2	sylib 207	. 2 ⊢ (𝜓 → ∀𝑥(𝜑 → ∀𝑥𝜑))
4		19.9ht 2128	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
5	3, 4	syl 17	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  ℲwnfOLD 1700

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-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nfOLD 1712

This theorem is referenced by:  19.9tOLD  2192