Theorem wl-nf2-nf 32464

Description: hba1 2137 is sufficient to let a df-nf 1701 style definition be stricter than nf5 2102. (Contributed by Wolf Lammen, 14-Sep-2021.)

Hypothesis

Ref	Expression
wl-nf2-nf.1	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Assertion

Ref	Expression
wl-nf2-nf	⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))

Proof of Theorem wl-nf2-nf

Step	Hyp	Ref	Expression
1		wl-nf2-nf.1	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2	1	imim2i 16	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥∀𝑥𝜑))
3		19.38 1757	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
4	2, 3	syl 17	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

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

This theorem is referenced by: (None)