Theorem rexcom13 2774

Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)

Assertion

Ref	Expression
rexcom13	⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C φ ↔ ∃z ∈ C ∃y ∈ B ∃x ∈ A φ)

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

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

Proof of Theorem rexcom13

Step	Hyp	Ref	Expression
1		rexcom 2773	. 2 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C φ ↔ ∃y ∈ B ∃x ∈ A ∃z ∈ C φ)
2		rexcom 2773	. . 3 ⊢ (∃x ∈ A ∃z ∈ C φ ↔ ∃z ∈ C ∃x ∈ A φ)
3	2	rexbii 2640	. 2 ⊢ (∃y ∈ B ∃x ∈ A ∃z ∈ C φ ↔ ∃y ∈ B ∃z ∈ C ∃x ∈ A φ)
4		rexcom 2773	. 2 ⊢ (∃y ∈ B ∃z ∈ C ∃x ∈ A φ ↔ ∃z ∈ C ∃y ∈ B ∃x ∈ A φ)
5	1, 3, 4	3bitri 262	1 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C φ ↔ ∃z ∈ C ∃y ∈ B ∃x ∈ A φ)

This proof depends on syntax axioms:   ↔ wb 176  ∃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:  rexrot4  2775