Theorem 19.23t 1116

Description: Closed form of Theorem 19.23 of [Margaris] p. 90.

Assertion

Ref	Expression
19.23t	|- (A.x(ps -> A.xps) -> (A.x(ph -> ps) <-> (E.xph -> ps)))

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		hba1 1003	. . 3 |- (A.x(ps -> A.xps) -> A.xA.x(ps -> A.xps))
2		ax-4 973	. . . . 5 |- (A.xps -> ps)
3		ax-4 973	. . . . 5 |- (A.x(ps -> A.xps) -> (ps -> A.xps))
4	2, 3	impbid2 518	. . . 4 |- (A.x(ps -> A.xps) -> (A.xps <-> ps))
5	4	imbi2d 612	. . 3 |- (A.x(ps -> A.xps) -> ((ph -> A.xps) <-> (ph -> ps)))
6	1, 5	albid 1104	. 2 |- (A.x(ps -> A.xps) -> (A.x(ph -> A.xps) <-> A.x(ph -> ps)))
7	4	imbi2d 612	. . 3 |- (A.x(ps -> A.xps) -> ((E.xph -> A.xps) <-> (E.xph -> ps)))
8		hba1 1003	. . . 4 |- (A.xps -> A.xA.xps)
9	8	19.23 1063	. . 3 |- (A.x(ph -> A.xps) <-> (E.xph -> A.xps))
10	7, 9	syl5bb 532	. 2 |- (A.x(ps -> A.xps) -> (A.x(ph -> A.xps) <-> (E.xph -> ps)))
11	6, 10	bitr3d 530	1 |- (A.x(ps -> A.xps) -> (A.x(ph -> ps) <-> (E.xph -> ps)))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954  E.wex 980

This theorem is referenced by:  vtoclegft 1856  sbciegft 1959

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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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