Theorem excom13 1743

Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
excom13	⊢ (∃x∃y∃zφ ↔ ∃z∃y∃xφ)

Proof of Theorem excom13

Step	Hyp	Ref	Expression
1		excom 1741	. 2 ⊢ (∃x∃y∃zφ ↔ ∃y∃x∃zφ)
2		excom 1741	. . 3 ⊢ (∃x∃zφ ↔ ∃z∃xφ)
3	2	exbii 1582	. 2 ⊢ (∃y∃x∃zφ ↔ ∃y∃z∃xφ)
4		excom 1741	. 2 ⊢ (∃y∃z∃xφ ↔ ∃z∃y∃xφ)
5	1, 3, 4	3bitri 262	1 ⊢ (∃x∃y∃zφ ↔ ∃z∃y∃xφ)

This proof depends on syntax axioms:   ↔ wb 176  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-7 1734

This proof depends on definitions:  df-bi 177  df-ex 1542

This theorem is used by:  exrot3  1744  exrot4  1745