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Theorem excom13 1595
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 1570 . 2 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑥𝑧𝜑)
2 excom 1570 . . 3 (∃𝑥𝑧𝜑 ↔ ∃𝑧𝑥𝜑)
32exbii 1512 . 2 (∃𝑦𝑥𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
4 excom 1570 . 2 (∃𝑦𝑧𝑥𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
51, 3, 43bitri 199 1 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  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-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  exrot3  1596  exrot4  1597  euotd  4018
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