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Theorem bj-19.12 34166
 Description: See 19.12 2347. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2169 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1786 or df-bj-nnf 34132, directly or indirectly. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-19.12 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem bj-19.12
StepHypRef Expression
1 bj-modalbe 34096 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑦𝑥𝑦𝜑)
2 excom 2169 . . 3 (∃𝑦𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝑦𝜑)
3 axc7e 2338 . . . 4 (∃𝑦𝑦𝜑𝜑)
43eximi 1836 . . 3 (∃𝑥𝑦𝑦𝜑 → ∃𝑥𝜑)
52, 4sylbi 220 . 2 (∃𝑦𝑥𝑦𝜑 → ∃𝑥𝜑)
61, 5sylg 1824 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  bj-nnflemae  34169  bj-nnflemea  34170
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