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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dvelimdv | Structured version Visualization version GIF version |
Description: Deduction form of dvelim 2325 with DV conditions. Typically, 𝑧 is a
fresh variable used for the implicit substitution hypothesis that
results in 𝜒 (namely, 𝜓 can be thought as 𝜓(𝑥, 𝑦) and
𝜒 as 𝜓(𝑥, 𝑧)). So the theorem says that if x is
effectively free in 𝜓(𝑥, 𝑧), then if x and y are not the same
variable, then 𝑥 is also effectively free in 𝜓(𝑥, 𝑦), in a
context 𝜑.
One can weakend the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use non-freeness hypotheses instead of DV conditions but since this result is typically used when 𝑧 is a dummy variable, this would not be of much benefit. One could also remove DV(z,x) since in the proof nfv 1830 can be replaced with nfal 2139 followed by nfn 1768. Remark: nfald 2151 uses ax-11 2021; it might be possible to inline and use ax11w 1994 instead, but there is still a use via 19.12 2150 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dvelimdv.nf | ⊢ Ⅎ𝑥𝜑 |
bj-dvelimdv.nf1 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
bj-dvelimdv.is | ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-dvelimdv | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-dvelimdv.is | . . . 4 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) | |
2 | 1 | equsalvw 1918 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓) |
3 | 2 | bicomi 213 | . 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜒)) |
4 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
5 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
6 | 4, 5 | nfan 1816 | . . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
7 | nfeqf2 2285 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
8 | 7 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦) |
9 | bj-dvelimdv.nf1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒) |
11 | 8, 10 | nfimd 1812 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒)) |
12 | 6, 11 | nfald 2151 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒)) |
13 | 3, 12 | nfxfrd 1772 | 1 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-axc14nf 32031 |
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