Theorem 19.12 1847

Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 1898 and r19.12sn 3790. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)

Assertion

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

Proof of Theorem 19.12

Step	Hyp	Ref	Expression
1		nfa1 1788	. . 3 ⊢ Ⅎy∀yφ
2	1	nfex 1843	. 2 ⊢ Ⅎy∃x∀yφ
3		sp 1747	. . 3 ⊢ (∀yφ → φ)
4	3	eximi 1576	. 2 ⊢ (∃x∀yφ → ∃xφ)
5	2, 4	alrimi 1765	1 ⊢ (∃x∀yφ → ∀y∃xφ)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12olem2  1928