Theorem 19.12 2329

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 2350 and r19.12sn 4611. (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 2156	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	nfex 2326	. 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑
3		sp 2184	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1841	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimi 2215	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfald  2330  bj-nfald  34929  pm11.61  41549