Theorem spc3gv 3298

Description: Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Hypothesis

Ref	Expression
spc3egv.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spc3gv	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem spc3gv

Step	Hyp	Ref	Expression
1		spc3egv.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
2	1	notbid 308	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	spc3egv 3297	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (¬ 𝜓 → ∃𝑥∃𝑦∃𝑧 ¬ 𝜑))
4		exnal 1754	. . . . . . 7 ⊢ (∃𝑧 ¬ 𝜑 ↔ ¬ ∀𝑧𝜑)
5	4	exbii 1774	. . . . . 6 ⊢ (∃𝑦∃𝑧 ¬ 𝜑 ↔ ∃𝑦 ¬ ∀𝑧𝜑)
6		exnal 1754	. . . . . 6 ⊢ (∃𝑦 ¬ ∀𝑧𝜑 ↔ ¬ ∀𝑦∀𝑧𝜑)
7	5, 6	bitri 264	. . . . 5 ⊢ (∃𝑦∃𝑧 ¬ 𝜑 ↔ ¬ ∀𝑦∀𝑧𝜑)
8	7	exbii 1774	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧 ¬ 𝜑 ↔ ∃𝑥 ¬ ∀𝑦∀𝑧𝜑)
9		exnal 1754	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦∀𝑧𝜑 ↔ ¬ ∀𝑥∀𝑦∀𝑧𝜑)
10	8, 9	bitr2i 265	. . 3 ⊢ (¬ ∀𝑥∀𝑦∀𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧 ¬ 𝜑)
11	3, 10	syl6ibr 242	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (¬ 𝜓 → ¬ ∀𝑥∀𝑦∀𝑧𝜑))
12	11	con4d 114	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  funopg  5922  pslem  17206  dirtr  17236  mclsax  31466  fununiq  31667