Theorem exrot4 2199

Description: Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
exrot4	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)

Proof of Theorem exrot4

Step	Hyp	Ref	Expression
1		excom13 2197	. . 3 ⊢ (∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑤∃𝑧∃𝑦𝜑)
2	1	exbii 1867	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑥∃𝑤∃𝑧∃𝑦𝜑)
3		excom13 2197	. 2 ⊢ (∃𝑥∃𝑤∃𝑧∃𝑦𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
4	2, 3	bitri 277	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-11 2190

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  elvvv  5719  dfoprab2  7449  xpassen  9037  5oalem7  31820  elfuns  36224  fundcmpsurbijinj  47977