Theorem nexdvOLD 1865

Description: Obsolete proof of nexdv 1864 as of 10-Oct-2021. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nexdv.1	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
nexdvOLD	⊢ (𝜑 → ¬ ∃𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem nexdvOLD

Step	Hyp	Ref	Expression
1		nexdv.1	. . 3 ⊢ (𝜑 → ¬ 𝜓)
2	1	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝜓)
3		alnex 1706	. 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
4	2, 3	sylib 208	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)