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Theorem bj-3exbi 33945
Description: Closed form of 3exbii 1846. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-3exbi (∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Proof of Theorem bj-3exbi
StepHypRef Expression
1 exbi 1843 . . 3 (∀𝑧(𝜑𝜓) → (∃𝑧𝜑 ↔ ∃𝑧𝜓))
212alimi 1809 . 2 (∀𝑥𝑦𝑧(𝜑𝜓) → ∀𝑥𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓))
3 bj-2exbi 33944 . 2 (∀𝑥𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))
42, 3syl 17 1 (∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 209  df-ex 1777
This theorem is referenced by: (None)
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