Theorem r2exf 2651

Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
r2alf.1	⊢ ℲyA

Assertion

Ref	Expression
r2exf	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))

Distinct variable group:   x,y

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

Proof of Theorem r2exf

Step	Hyp	Ref	Expression
1		df-rex 2621	. 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x(x ∈ A ∧ ∃y ∈ B φ))
2		r2alf.1	. . . . . 6 ⊢ ℲyA
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎy x ∈ A
4	3	19.42 1880	. . . 4 ⊢ (∃y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
5		anass 630	. . . . 5 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) ↔ (x ∈ A ∧ (y ∈ B ∧ φ)))
6	5	exbii 1582	. . . 4 ⊢ (∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃y(x ∈ A ∧ (y ∈ B ∧ φ)))
7		df-rex 2621	. . . . 5 ⊢ (∃y ∈ B φ ↔ ∃y(y ∈ B ∧ φ))
8	7	anbi2i 675	. . . 4 ⊢ ((x ∈ A ∧ ∃y ∈ B φ) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
9	4, 6, 8	3bitr4i 268	. . 3 ⊢ (∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ (x ∈ A ∧ ∃y ∈ B φ))
10	9	exbii 1582	. 2 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃x(x ∈ A ∧ ∃y ∈ B φ))
11	1, 10	bitr4i 243	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Ⅎwnfc 2477  ∃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-or 359  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-rex 2621

This theorem is used by:  r2ex  2653  rexcomf  2771