Theorem bj-dvelimdv 32834

Description: Deduction form of dvelim 2337 with DV conditions. Uncurried (imported) form of bj-dvelimdv 32834. 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 1843 can be replaced with nfal 2153 followed by nfn 1784.

Remark: nfald 2165 uses ax-11 2034; it might be possible to inline and use ax11w 2007 instead, but there is still a use via 19.12 2164 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-dvelimdv.nf	⊢ Ⅎ𝑥𝜑
bj-dvelimdv.nf1	⊢ (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 1931	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓)
3	2	bicomi 214	. 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜒))
4		nfv 1843	. . . 4 ⊢ Ⅎ𝑧𝜑
5		nfv 1843	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6	4, 5	nfan 1828	. . 3 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
7		nfeqf2 2297	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
8	7	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
9		bj-dvelimdv.nf1	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
11	8, 10	nfimd 1823	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))
12	6, 11	nfald 2165	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒))
13	3, 12	nfxfrd 1780	1 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)