Theorem r19.12 1281

Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers.

Assertion

Ref	Expression
r19.12	⊢ (∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A φ)

Distinct variable group(s):   x,y   y,A   x,B

Proof of Theorem r19.12

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
2		hbra1 1237	. . . 4 ⊢ (∀y ∈ B φ → ∀y∀y ∈ B φ)
3	1, 2	hbrex 1238	. . 3 ⊢ (∃x ∈ A ∀y ∈ B φ → ∀y∃x ∈ A ∀y ∈ B φ)
4		ax-1 3	. . 3 ⊢ (∃x ∈ A ∀y ∈ B φ → (y ∈ B → ∃x ∈ A ∀y ∈ B φ))
5	3, 4	r19.21ai 1258	. 2 ⊢ (∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A ∀y ∈ B φ)
6		ra4 1243	. . . . . 6 ⊢ (∀y ∈ B φ → (y ∈ B → φ))
7	6	com12 13	. . . . 5 ⊢ (y ∈ B → (∀y ∈ B φ → φ))
8	7	a1d 14	. . . 4 ⊢ (y ∈ B → (x ∈ A → (∀y ∈ B φ → φ)))
9	8	r19.22dv 1278	. . 3 ⊢ (y ∈ B → (∃x ∈ A ∀y ∈ B φ → ∃x ∈ A φ))
10	9	r19.20i 1253	. 2 ⊢ (∀y ∈ B ∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A φ)
11	5, 10	syl 12	1 ⊢ (∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A φ)

Syntax hints:   → wi 2   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  iuniin 2001

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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