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Mirrors > Home > ILE Home > Th. List > nf4dc | GIF version |
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 1664, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Ref | Expression |
---|---|
nf4dc | ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf2 1661 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | imordc 892 | . . 3 ⊢ (DECID ∃𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))) | |
3 | 1, 2 | syl5bb 191 | . 2 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))) |
4 | orcom 723 | . . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | |
5 | alnex 1492 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | 5 | orbi2i 757 | . . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) |
7 | 4, 6 | bitr4i 186 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
8 | 3, 7 | bitrdi 195 | 1 ⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 703 DECID wdc 829 ∀wal 1346 Ⅎwnf 1453 ∃wex 1485 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-gen 1442 ax-ie2 1487 ax-4 1503 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-fal 1354 df-nf 1454 |
This theorem is referenced by: (None) |
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