Theorem excom13 2166

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 2164	. 2 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑥∃𝑧𝜑)
2		excom 2164	. . 3 ⊢ (∃𝑥∃𝑧𝜑 ↔ ∃𝑧∃𝑥𝜑)
3	2	exbii 1851	. 2 ⊢ (∃𝑦∃𝑥∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
4		excom 2164	. 2 ⊢ (∃𝑦∃𝑧∃𝑥𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
5	1, 3, 4	3bitri 296	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  exrot3  2167  exrot4  2168  euotd  5421  elfuns  34144  fundcmpsurbijinj  44750