Theorem ee8anv 1923

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 1679	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓))
2	1	2exbii 1594	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓))
3		ee4anv 1922	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
4	3	2exbii 1594	. . 3 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
5	4	2exbii 1594	. 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓))
6		ee4anv 1922	. 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))
7	2, 5, 6	3bitri 205	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  enq0tr  7375