Theorem 2reu5lem2 3043

Description: Lemma for 2reu5 3045. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5lem2	⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ))

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

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

Proof of Theorem 2reu5lem2

Step	Hyp	Ref	Expression
1		df-rmo 2623	. . 3 ⊢ (∃*y ∈ B φ ↔ ∃*y(y ∈ B ∧ φ))
2	1	ralbii 2639	. 2 ⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x ∈ A ∃*y(y ∈ B ∧ φ))
3		df-ral 2620	. . 3 ⊢ (∀x ∈ A ∃*y(y ∈ B ∧ φ) ↔ ∀x(x ∈ A → ∃*y(y ∈ B ∧ φ)))
4		moanimv 2262	. . . . . 6 ⊢ (∃*y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A → ∃*y(y ∈ B ∧ φ)))
5	4	bicomi 193	. . . . 5 ⊢ ((x ∈ A → ∃*y(y ∈ B ∧ φ)) ↔ ∃*y(x ∈ A ∧ (y ∈ B ∧ φ)))
6		3anass 938	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ B ∧ φ) ↔ (x ∈ A ∧ (y ∈ B ∧ φ)))
7	6	bicomi 193	. . . . . 6 ⊢ ((x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ y ∈ B ∧ φ))
8	7	mobii 2240	. . . . 5 ⊢ (∃*y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ ∃*y(x ∈ A ∧ y ∈ B ∧ φ))
9	5, 8	bitri 240	. . . 4 ⊢ ((x ∈ A → ∃*y(y ∈ B ∧ φ)) ↔ ∃*y(x ∈ A ∧ y ∈ B ∧ φ))
10	9	albii 1566	. . 3 ⊢ (∀x(x ∈ A → ∃*y(y ∈ B ∧ φ)) ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ))
11	3, 10	bitri 240	. 2 ⊢ (∀x ∈ A ∃*y(y ∈ B ∧ φ) ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ))
12	2, 11	bitri 240	1 ⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615  ∃*wrmo 2618

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

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2620  df-rmo 2623

This theorem is used by:  2reu5lem3  3044