Theorem 19.12 1047

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 1302 and r19.12sn 2442.

Assertion

Ref	Expression
19.12	|- (E.xA.yph -> A.yE.xph)

Proof of Theorem 19.12

Step	Hyp	Ref	Expression
1		hba1 1003	. . 3 |- (A.yph -> A.yA.yph)
2	1	hbex 1006	. 2 |- (E.xA.yph -> A.yE.xA.yph)
3		ax-4 973	. . 3 |- (A.yph -> ph)
4	3	19.22i 1040	. 2 |- (E.xA.yph -> E.xph)
5	2, 4	19.21ai 998	1 |- (E.xA.yph -> A.yE.xph)

Syntax hints:   -> wi 3  A.wal 954  E.wex 980

This theorem is referenced by:  hbexd 1114  iinss 2598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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