Theorem 19.12 1713

Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)

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		hba1 1589	. . 3 |- ( A. y ph -> A. y A. y ph )
2	1	hbex 1685	. 2 |- ( E. x A. y ph -> A. y E. x A. y ph )
3		ax-4 1559	. . 3 |- ( A. y ph -> ph )
4	3	eximi 1649	. 2 |- ( E. x A. y ph -> E. x ph )
5	2, 4	alrimih 1518	1 |- ( E. x A. y ph -> A. y E. x ph )

Syntax hints:   -> wi 4   A.wal 1396   E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbexd  1742  nfexd  1809  cbvexdh  1975