Theorem nf4dc 1684

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 1685, 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 1682	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		imordc 898	. . 3 ⊢ (DECID ∃𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
3	1, 2	bitrid 192	. 2 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
4		orcom 729	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
5		alnex 1513	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	5	orbi2i 763	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
7	4, 6	bitr4i 187	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
8	3, 7	bitrdi 196	1 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709  DECID wdc 835  ∀wal 1362  Ⅎwnf 1474  ∃wex 1506

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-gen 1463  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1475

This theorem is referenced by: (None)