Theorem bj-19.12 36744

Description: See 19.12 2326. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2160 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1781 or df-bj-nnf 36707, 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 36671	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑)
2		excom 2160	. . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑)
3		axc7e 2317	. . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑)
4	3	eximi 1832	. . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	sylbi 217	. 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
6	1, 5	sylg 1820	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by:  bj-nnflemae  36747  bj-nnflemea  36748