Theorem bj-19.12 36937

Description: See 19.12 2333. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2168 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 36900, directly or indirectly. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-19.12	⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Proof of Theorem bj-19.12

Step	Hyp	Ref	Expression
1		bj-modalbe 36864	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑)
2		excom 2168	. . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑)
3		axc7e 2324	. . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑)
4	3	eximi 1837	. . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	sylbi 217	. 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
6	1, 5	sylg 1825	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  bj-nnflemae  36940  bj-nnflemea  36941