Theorem ee8anv 1810

Description: Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)

Assertion

Ref	Expression
ee8anv	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))

Distinct variable groups:   𝜑,𝑣   𝜑,𝑢   𝜑,𝑡   𝜑,𝑠   𝜓,𝑥   𝜓,𝑦   𝜓,𝑧   𝜓,𝑤   𝑥,𝑠   𝑦,𝑠   𝑧,𝑠   𝑤,𝑡   𝑥,𝑡   𝑦,𝑡   𝑤,𝑢   𝑥,𝑢   𝑧,𝑢   𝑤,𝑣   𝑦,𝑣   𝑧,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑣,𝑢,𝑡,𝑠)

Proof of Theorem ee8anv

Step	Hyp	Ref	Expression
1		exrot4 1581	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓))
2	1	2exbii 1497	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓))
3		ee4anv 1809	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
4	3	2exbii 1497	. . 3 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
5	4	2exbii 1497	. 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
6		ee4anv 1809	. 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))
7	2, 5, 6	3bitri 195	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  enq0tr  6532