Theorem excom13 776

Description: Swap 1st and 3rd existential quantifiers.

Assertion

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

Proof of Theorem excom13

Step	Hyp	Ref	Expression
1		excom 728	. 2 ⊢ (∃x∃y∃zφ ↔ ∃y∃x∃zφ)
2		excom 728	. . 3 ⊢ (∃x∃zφ ↔ ∃z∃xφ)
3	2	biex 733	. 2 ⊢ (∃y∃x∃zφ ↔ ∃y∃z∃xφ)
4		excom 728	. 2 ⊢ (∃y∃z∃xφ ↔ ∃z∃y∃xφ)
5	1, 3, 4	3bitr 155	1 ⊢ (∃x∃y∃zφ ↔ ∃z∃y∃xφ)

Syntax hints:   ↔ wb 127  ∃wex 678

This theorem is referenced by:  exrot3 777  exrot4 778

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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