Theorem bj-dvelimdv 36817

Description: Deduction form of dvelim 2459 with disjoint variable conditions. Uncurried (imported) form of bj-dvelimdv1 36818. 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 1913 can be replaced with nfal 2327 followed by nfn 1856.

Remark: nfald 2332 uses ax-11 2158; it might be possible to inline and use ax11w 2130 instead, but there is still a use via 19.12 2331 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 2003	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓)
3	2	bicomi 224	. 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜒))
4		nfv 1913	. . . 4 ⊢ Ⅎ𝑧𝜑
5		nfv 1913	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6	4, 5	nfan 1898	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
7		nfeqf2 2385	. . . . 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 1893	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))
12	6, 11	nfald 2332	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒))
13	3, 12	nfxfrd 1852	1 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782

This theorem is referenced by: (None)