Theorem bj-19.12 37464

Description: See 19.12 2359. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2199 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1817 or df-bj-nnf 37468, 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 37429	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑)
2		excom 2199	. . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑)
3		axc7e 2350	. . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑)
4	3	eximi 1868	. . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	sylbi 220	. 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
6	1, 5	sylg 1856	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-nnflemae  37529  bj-nnflemea  37530