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Theorem excom13 1579
Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
excom13 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Proof of Theorem excom13
StepHypRef Expression
1 excom 1554 . 2 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑥𝑧𝜑)
2 excom 1554 . . 3 (∃𝑥𝑧𝜑 ↔ ∃𝑧𝑥𝜑)
32exbii 1496 . 2 (∃𝑦𝑥𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
4 excom 1554 . 2 (∃𝑦𝑧𝑥𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
51, 3, 43bitri 195 1 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  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-ial 1427
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  exrot3  1580  exrot4  1581  euotd  3991
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