Theorem 19.23 745

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

Hypothesis

Ref	Expression
19.23.1	|- (ps -> A.xps)

Assertion

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

Proof of Theorem 19.23

Step	Hyp	Ref	Expression
1		19.22 722	. . 3 |- (A.x(ph -> ps) -> (E.xph -> E.xps))
2		19.23.1	. . . 4 |- (ps -> A.xps)
3	2	19.9r 718	. . 3 |- (ps <-> E.xps)
4	1, 3	syl6ibr 186	. 2 |- (A.x(ph -> ps) -> (E.xph -> ps))
5		hbe1 709	. . . 4 |- (E.xph -> A.xE.xph)
6	5, 2	hbim 702	. . 3 |- ((E.xph -> ps) -> A.x(E.xph -> ps))
7		19.8a 712	. . . 4 |- (ph -> E.xph)
8	7	syl4 19	. . 3 |- ((E.xph -> ps) -> (ph -> ps))
9	6, 8	19.21ai 740	. 2 |- ((E.xph -> ps) -> A.x(ph -> ps))
10	4, 9	impbi 139	1 |- (A.x(ph -> ps) <-> (E.xph -> ps))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678

This theorem is referenced by:  19.23ad 748  sbied 903  19.23v 950  ceqsalg 1362  ralidm 1774

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org