Theorem r19.12 2728

Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Distinct variable groups:   x,y   y,A   x,B

Allowed substitution hints:   φ(x, y)   A(x)   B(y)

Proof of Theorem r19.12

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 ⊢ ℲyA
2		nfra1 2665	. . . 4 ⊢ Ⅎy∀y ∈ B φ
3	1, 2	nfrex 2670	. . 3 ⊢ Ⅎy∃x ∈ A ∀y ∈ B φ
4		ax-1 6	. . 3 ⊢ (∃x ∈ A ∀y ∈ B φ → (y ∈ B → ∃x ∈ A ∀y ∈ B φ))
5	3, 4	ralrimi 2696	. 2 ⊢ (∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A ∀y ∈ B φ)
6		rsp 2675	. . . . 5 ⊢ (∀y ∈ B φ → (y ∈ B → φ))
7	6	com12 27	. . . 4 ⊢ (y ∈ B → (∀y ∈ B φ → φ))
8	7	reximdv 2726	. . 3 ⊢ (y ∈ B → (∃x ∈ A ∀y ∈ B φ → ∃x ∈ A φ))
9	8	ralimia 2688	. 2 ⊢ (∀y ∈ B ∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A φ)
10	5, 9	syl 15	1 ⊢ (∃x ∈ A ∀y ∈ B φ → ∀y ∈ B ∃x ∈ A φ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This proof depends on 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-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621

This theorem is used by:  iuniin  3980