Theorem cla43egv 1869

Description: Existential specialization with 3 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla43egv.1	⊢ ((x = A ⋀ y = B ⋀ z = C) → (φ ↔ ψ))

Assertion

Ref	Expression
cla43egv	⊢ ((A ∈ R ⋀ B ∈ S ⋀ C ∈ T) → (ψ → ∃x∃y∃zφ))

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

Proof of Theorem cla43egv

Step	Hyp	Ref	Expression
1		cla43egv.1	. . . . 5 ⊢ ((x = A ⋀ y = B ⋀ z = C) → (φ ↔ ψ))
2	1	biimprcd 156	. . . 4 ⊢ (ψ → ((x = A ⋀ y = B ⋀ z = C) → φ))
3	2	19.22dv 1292	. . 3 ⊢ (ψ → (∃z(x = A ⋀ y = B ⋀ z = C) → ∃zφ))
4	3	19.22dvv 1294	. 2 ⊢ (ψ → (∃x∃y∃z(x = A ⋀ y = B ⋀ z = C) → ∃x∃y∃zφ))
5		elex 1822	. . . 4 ⊢ (A ∈ R → ∃x x = A)
6		elex 1822	. . . 4 ⊢ (B ∈ S → ∃y y = B)
7		elex 1822	. . . 4 ⊢ (C ∈ T → ∃z z = C)
8	5, 6, 7	3anim123i 823	. . 3 ⊢ ((A ∈ R ⋀ B ∈ S ⋀ C ∈ T) → (∃x x = A ⋀ ∃y y = B ⋀ ∃z z = C))
9		eeeanv 1326	. . 3 ⊢ (∃x∃y∃z(x = A ⋀ y = B ⋀ z = C) ↔ (∃x x = A ⋀ ∃y y = B ⋀ ∃z z = C))
10	8, 9	sylibr 200	. 2 ⊢ ((A ∈ R ⋀ B ∈ S ⋀ C ∈ T) → ∃x∃y∃z(x = A ⋀ y = B ⋀ z = C))
11	4, 10	syl5com 52	1 ⊢ ((A ∈ R ⋀ B ∈ S ⋀ C ∈ T) → (ψ → ∃x∃y∃zφ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∃wex 982

This theorem is referenced by:  cla43gv 1870

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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