Theorem exrot4 2203

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 2201	. . 3 ⊢ (∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑤∃𝑧∃𝑦𝜑)
2	1	exbii 1881	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑥∃𝑤∃𝑧∃𝑦𝜑)
3		excom13 2201	. 2 ⊢ (∃𝑥∃𝑤∃𝑧∃𝑦𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-11 2194

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  elvvv  5727  dfoprab2  7476  xpassen  9083  5oalem7  32255  elfuns  36657  fundcmpsurbijinj  48461