Theorem bj-dvelimdv 37341

Description: Deduction form of dvelim 2484 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 37342. 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 1936 can be replaced with nfal 2357 followed by nfn 1879.

Remark: nfald 2362 uses ax-11 2193; it might be possible to inline and use ax11w 2166 instead, but there is still a use via 19.12 2361 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 2026	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓)
3	2	bicomi 226	. 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜒))
4		nfv 1936	. . . 4 ⊢ Ⅎ𝑧𝜑
5		nfv 1936	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6	4, 5	nfan 1921	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
7		nfeqf2 2410	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
8	7	adantl 485	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
9		bj-dvelimdv.nf	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
11	8, 10	nfimd 1916	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))
12	6, 11	nfald 2362	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒))
13	3, 12	nfxfrd 1876	1 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560  Ⅎwnf 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806

This theorem is referenced by: (None)