Theorem nf4dc 1648

Description: Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1649, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)

Assertion

Ref	Expression
nf4dc	⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Proof of Theorem nf4dc

Step	Hyp	Ref	Expression
1		nf2 1646	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		imordc 882	. . 3 ⊢ (DECID ∃𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
3	1, 2	syl5bb 191	. 2 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
4		orcom 717	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
5		alnex 1475	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	5	orbi2i 751	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
7	4, 6	bitr4i 186	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
8	3, 7	syl6bb 195	1 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 697  DECID wdc 819  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by: (None)