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Theorem bj-19.12 36762
Description: See 19.12 2327. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2162 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1784 or df-bj-nnf 36725, 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 36689 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑦𝑥𝑦𝜑)
2 excom 2162 . . 3 (∃𝑦𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝑦𝜑)
3 axc7e 2318 . . . 4 (∃𝑦𝑦𝜑𝜑)
43eximi 1835 . . 3 (∃𝑥𝑦𝑦𝜑 → ∃𝑥𝜑)
52, 4sylbi 217 . 2 (∃𝑦𝑥𝑦𝜑 → ∃𝑥𝜑)
61, 5sylg 1823 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-ex 1780
This theorem is referenced by:  bj-nnflemae  36765  bj-nnflemea  36766
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