Theorem exbi 1397

Description: Theorem 19.18 of [Margaris] p. 90.

Assertion

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

Proof of Theorem exbi

Step	Hyp	Ref	Expression
1		bi1 165	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
2	1	alimi 1338	. . 3 |- (A.x(ph <-> ps) -> A.x(ph -> ps))
3		exim 1386	. . 3 |- (A.x(ph -> ps) -> (E.xph -> E.xps))
4	2, 3	syl 12	. 2 |- (A.x(ph <-> ps) -> (E.xph -> E.xps))
5		bi2 166	. . . 4 |- ((ph <-> ps) -> (ps -> ph))
6	5	alimi 1338	. . 3 |- (A.x(ph <-> ps) -> A.x(ps -> ph))
7		exim 1386	. . 3 |- (A.x(ps -> ph) -> (E.xps -> E.xph))
8	6, 7	syl 12	. 2 |- (A.x(ph <-> ps) -> (E.xps -> E.xph))
9	4, 8	impbid 574	1 |- (A.x(ph <-> ps) -> (E.xph <-> E.xps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296  E.wex 1326

This theorem is referenced by:  exbii 1398  19.19 1402  exbid 1460  exintrbi 1476  bnj957 12852  bnj1157 12949  2exbi 16332  rexbidar 16423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

Copyright terms: Public domain