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Mirrors > Home > ILE Home > Th. List > nf4r | GIF version |
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 1681. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Ref | Expression |
---|---|
nf4r | ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 729 | . . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | |
2 | alnex 1510 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
3 | 2 | orbi2i 763 | . . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) |
4 | 1, 3 | bitr4i 187 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
5 | imorr 722 | . . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) | |
6 | nf2 1679 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
7 | 5, 6 | sylibr 134 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
8 | 4, 7 | sylbir 135 | 1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 709 ∀wal 1362 Ⅎwnf 1471 ∃wex 1503 |
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 1458 ax-gen 1460 ax-ie2 1505 ax-4 1521 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 |
This theorem is referenced by: (None) |
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