Theorem 19.23 1059

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 1035	. . 3 |- (A.x(ph -> ps) -> (E.xph -> E.xps))
2		19.23.1	. . . 4 |- (ps -> A.xps)
3	2	19.9 1032	. . 3 |- (E.xps <-> ps)
4	1, 3	syl6ib 212	. 2 |- (A.x(ph -> ps) -> (E.xph -> ps))
5		hbe1 1012	. . . 4 |- (E.xph -> A.xE.xph)
6	5, 2	hbim 1004	. . 3 |- ((E.xph -> ps) -> A.x(E.xph -> ps))
7		19.8a 1025	. . . 4 |- (ph -> E.xph)
8	7	imim1i 16	. . 3 |- ((E.xph -> ps) -> (ph -> ps))
9	6, 8	19.21ai 995	. 2 |- ((E.xph -> ps) -> A.x(ph -> ps))
10	4, 9	impbi 157	1 |- (A.x(ph -> ps) <-> (E.xph -> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951  E.wex 977

This theorem is referenced by:  19.23ad 1062  19.23t 1112  sbied 1191  19.23v 1288  ceqsalg 1816  ralidm 2347  r19.3rzvb 10337

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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