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Theorem excom13 2084
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 2082 . 2 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑥𝑧𝜑)
2 excom 2082 . . 3 (∃𝑥𝑧𝜑 ↔ ∃𝑧𝑥𝜑)
32exbii 1814 . 2 (∃𝑦𝑥𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
4 excom 2082 . 2 (∃𝑦𝑧𝑥𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
51, 3, 43bitri 286 1 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-11 2074
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  exrot3  2085  exrot4  2086  euotd  5004  elfuns  32147
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