Theorem excom13 2170

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 2168	. 2 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑥∃𝑧𝜑)
2		excom 2168	. . 3 ⊢ (∃𝑥∃𝑧𝜑 ↔ ∃𝑧∃𝑥𝜑)
3	2	exbii 1855	. 2 ⊢ (∃𝑦∃𝑥∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
4		excom 2168	. 2 ⊢ (∃𝑦∃𝑧∃𝑥𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
5	1, 3, 4	3bitri 300	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-11 2160

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  exrot3  2171  exrot4  2172  euotd  5381  elfuns  33903  fundcmpsurbijinj  44478