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Theorem ee8anv 1826
Description: Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
ee8anv (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝑤𝜑 ∧ ∃𝑣𝑢𝑡𝑠𝜓))
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
StepHypRef Expression
1 exrot4 1597 . . 3 (∃𝑧𝑤𝑣𝑢𝑡𝑠(𝜑𝜓) ↔ ∃𝑣𝑢𝑧𝑤𝑡𝑠(𝜑𝜓))
212exbii 1513 . 2 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(𝜑𝜓) ↔ ∃𝑥𝑦𝑣𝑢𝑧𝑤𝑡𝑠(𝜑𝜓))
3 ee4anv 1825 . . . 4 (∃𝑧𝑤𝑡𝑠(𝜑𝜓) ↔ (∃𝑧𝑤𝜑 ∧ ∃𝑡𝑠𝜓))
432exbii 1513 . . 3 (∃𝑣𝑢𝑧𝑤𝑡𝑠(𝜑𝜓) ↔ ∃𝑣𝑢(∃𝑧𝑤𝜑 ∧ ∃𝑡𝑠𝜓))
542exbii 1513 . 2 (∃𝑥𝑦𝑣𝑢𝑧𝑤𝑡𝑠(𝜑𝜓) ↔ ∃𝑥𝑦𝑣𝑢(∃𝑧𝑤𝜑 ∧ ∃𝑡𝑠𝜓))
6 ee4anv 1825 . 2 (∃𝑥𝑦𝑣𝑢(∃𝑧𝑤𝜑 ∧ ∃𝑡𝑠𝜓) ↔ (∃𝑥𝑦𝑧𝑤𝜑 ∧ ∃𝑣𝑢𝑡𝑠𝜓))
72, 5, 63bitri 199 1 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝑤𝜑 ∧ ∃𝑣𝑢𝑡𝑠𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  enq0tr  6590
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