Theorem nf4r 1613

Description: If 𝜑 is always true or always false, then variable 𝑥 is effectively not free in 𝜑. The converse holds given a decidability condition, as seen at nf4dc 1612. (Contributed by Jim Kingdon, 21-Jul-2018.)

Assertion

Ref	Expression
nf4r	⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem nf4r

Step	Hyp	Ref	Expression
1		orcom 685	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1440	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	orbi2i 717	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 3	bitr4i 186	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
5		imorr 838	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
6		nf2 1610	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7	5, 6	sylibr 133	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → Ⅎ𝑥𝜑)
8	4, 7	sylbir 134	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 667  ∀wal 1294  Ⅎwnf 1401  ∃wex 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-gen 1390  ax-ie2 1435  ax-4 1452  ax-ial 1479

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402

This theorem is referenced by: (None)