Theorem 19.12 2357

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 2376 and r19.12sn 4677. (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 2187	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2354	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2219	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1867	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2249	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  nfald  2358  bj-hbexd  37736  bj-hbex  37738  bj-nfald  38193  pm11.61  45514