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Theorem excom13 2031
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 2029 . 2 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑥𝑧𝜑)
2 excom 2029 . . 3 (∃𝑥𝑧𝜑 ↔ ∃𝑧𝑥𝜑)
32exbii 1764 . 2 (∃𝑦𝑥𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
4 excom 2029 . 2 (∃𝑦𝑧𝑥𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
51, 3, 43bitri 285 1 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-11 2021
This theorem depends on definitions:  df-bi 196  df-ex 1696
This theorem is referenced by:  exrot3  2032  exrot4  2033  euotd  4900  elfuns  31192
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