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Theorem bj-19.12 35255
Description: See 19.12 2321. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2163 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1787 or df-bj-nnf 35218, 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 35182 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑦𝑥𝑦𝜑)
2 excom 2163 . . 3 (∃𝑦𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝑦𝜑)
3 axc7e 2312 . . . 4 (∃𝑦𝑦𝜑𝜑)
43eximi 1838 . . 3 (∃𝑥𝑦𝑦𝜑 → ∃𝑥𝜑)
52, 4sylbi 216 . 2 (∃𝑦𝑥𝑦𝜑 → ∃𝑥𝜑)
61, 5sylg 1826 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  bj-nnflemae  35258  bj-nnflemea  35259
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