Theorem 19.12 2359

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 2378 and r19.12sn 4679. (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 2185	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2356	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2218	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1855	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2248	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by:  nfald  2360  bj-hbexd  37185  bj-hbex  37187  bj-nfald  37627  pm11.61  44969