Theorem 19.12 2360

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 2379 and r19.12sn 4686. (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 2186	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2357	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2219	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1865	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2249	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by:  nfald  2361  bj-hbexd  37355  bj-hbex  37357  bj-nfald  37799  pm11.61  45123