Theorem 19.12 2321

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 2345 and r19.12sn 4656. (Contributed by NM, 12-Mar-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 2148	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2318	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2176	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1837	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2206	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  nfald  2322  bj-nfald  35308  pm11.61  42011