Theorem 19.12 2164

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 2180 and r19.12sn 4255. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)

Assertion

Ref	Expression
19.12	|- ( E. x A. y ph -> A. y E. x ph )

Proof of Theorem 19.12

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ y A. y ph
2	1	nfex 2154	. 2 |- F/ y E. x A. y ph
3		sp 2053	. . 3 |- ( A. y ph -> ph )
4	3	eximi 1762	. 2 |- ( E. x A. y ph -> E. x ph )
5	2, 4	alrimi 2082	1 |- ( E. x A. y ph -> A. y E. x ph )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfald  2165  pm11.61  38593