Theorem rexrot4 2775

Description: Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)

Assertion

Ref	Expression
rexrot4	⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C ∃w ∈ D φ ↔ ∃z ∈ C ∃w ∈ D ∃x ∈ A ∃y ∈ B φ)

Distinct variable groups:   z,w,A   w,B,z   x,w,y,C   x,z,D,y

Allowed substitution hints:   φ(x,y,z,w)   A(x,y)   B(x,y)   C(z)   D(w)

Proof of Theorem rexrot4

Step	Hyp	Ref	Expression
1		rexcom13 2774	. . 3 ⊢ (∃y ∈ B ∃z ∈ C ∃w ∈ D φ ↔ ∃w ∈ D ∃z ∈ C ∃y ∈ B φ)
2	1	rexbii 2640	. 2 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C ∃w ∈ D φ ↔ ∃x ∈ A ∃w ∈ D ∃z ∈ C ∃y ∈ B φ)
3		rexcom13 2774	. 2 ⊢ (∃x ∈ A ∃w ∈ D ∃z ∈ C ∃y ∈ B φ ↔ ∃z ∈ C ∃w ∈ D ∃x ∈ A ∃y ∈ B φ)
4	2, 3	bitri 240	1 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C ∃w ∈ D φ ↔ ∃z ∈ C ∃w ∈ D ∃x ∈ A ∃y ∈ B φ)

Syntax hints:   ↔ wb 176  ∃wrex 2616

This theorem was proved from 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 theorem 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 referenced by: (None)