Theorem nf4r 1719

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 1718. (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 735	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1547	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	orbi2i 769	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 3	bitr4i 187	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
5		imorr 728	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
6		nf2 1716	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7	5, 6	sylibr 134	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → Ⅎ𝑥𝜑)
8	4, 7	sylbir 135	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 715  ∀wal 1395  Ⅎwnf 1508  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-gen 1497  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509

This theorem is referenced by: (None)