Theorem bj-19.12 36774

Description: See 19.12 2327. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2164 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1785 or df-bj-nnf 36737, 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 36701	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑)
2		excom 2164	. . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑)
3		axc7e 2318	. . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑)
4	3	eximi 1836	. . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	sylbi 217	. 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
6	1, 5	sylg 1824	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2143  ax-11 2159  ax-12 2179

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  bj-nnflemae  36777  bj-nnflemea  36778