Theorem exrot4 1702

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 1700	. . 3 ⊢ (∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑤∃𝑧∃𝑦𝜑)
2	1	exbii 1616	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑥∃𝑤∃𝑧∃𝑦𝜑)
3		excom13 1700	. 2 ⊢ (∃𝑥∃𝑤∃𝑧∃𝑦𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
4	2, 3	bitri 184	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)

Syntax hints:   ↔ wb 105  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ee8anv  1951  elvvv  4722  dfoprab2  5965  xpassen  6884  enq0sym  7492