Theorem 19.12 2326

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 2348 and r19.12sn 4725. (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 2149	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2323	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2181	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1832	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2211	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-or 848  df-ex 1777  df-nf 1781

This theorem is referenced by:  nfald  2327  bj-nfald  37120  pm11.61  44389