Theorem bj-19.12 36727

Description: See 19.12 2331. 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 1782 or df-bj-nnf 36690, 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 36654	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑)
2		excom 2163	. . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑)
3		axc7e 2322	. . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑)
4	3	eximi 1833	. . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	sylbi 217	. 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
6	1, 5	sylg 1821	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778

This theorem is referenced by:  bj-nnflemae  36730  bj-nnflemea  36731