Theorem bj-dvelimdv 36895

Description: Deduction form of dvelim 2451 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 36896. 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 weaken the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use nonfreeness hypotheses instead of disjoint variable 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 (𝑥, 𝑧) since in the proof nfv 1915 can be replaced with nfal 2324 followed by nfn 1858.

Remark: nfald 2329 uses ax-11 2160; it might be possible to inline and use ax11w 2133 instead, but there is still a use via 19.12 2328 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-dvelimdv.nf	⊢ (𝜑 → Ⅎ𝑥𝜒)
bj-dvelimdv.is	⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
bj-dvelimdv	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧   𝜓,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem bj-dvelimdv

Step	Hyp	Ref	Expression
1		bj-dvelimdv.is	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓))
2	1	equsalvw 2005	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓)
3	2	bicomi 224	. 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜒))
4		nfv 1915	. . . 4 ⊢ Ⅎ𝑧𝜑
5		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6	4, 5	nfan 1900	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
7		nfeqf2 2377	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
8	7	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
9		bj-dvelimdv.nf	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
11	8, 10	nfimd 1895	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))
12	6, 11	nfald 2329	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒))
13	3, 12	nfxfrd 1855	1 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)