Theorem spc3egv 2944

Description: Existential specialization with 3 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
spc3egv	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (ψ → ∃x∃y∃zφ))

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 spc3egv

Step	Hyp	Ref	Expression
1		elisset 2870	. . . 4 ⊢ (A ∈ V → ∃x x = A)
2		elisset 2870	. . . 4 ⊢ (B ∈ W → ∃y y = B)
3		elisset 2870	. . . 4 ⊢ (C ∈ X → ∃z z = C)
4	1, 2, 3	3anim123i 1137	. . 3 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
5		eeeanv 1914	. . 3 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
6	4, 5	sylibr 203	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → ∃x∃y∃z(x = A ∧ y = B ∧ z = C))
7		spc3egv.1	. . . . 5 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
8	7	biimprcd 216	. . . 4 ⊢ (ψ → ((x = A ∧ y = B ∧ z = C) → φ))
9	8	eximdv 1622	. . 3 ⊢ (ψ → (∃z(x = A ∧ y = B ∧ z = C) → ∃zφ))
10	9	2eximdv 1624	. 2 ⊢ (ψ → (∃x∃y∃z(x = A ∧ y = B ∧ z = C) → ∃x∃y∃zφ))
11	6, 10	syl5com 26	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (ψ → ∃x∃y∃zφ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  spc3gv  2945