Theorem 19.18 732

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

Assertion

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

Proof of Theorem 19.18

Step	Hyp	Ref	Expression
1		bi1 130	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
2	1	19.20i 691	. . 3 |- (A.x(ph <-> ps) -> A.x(ph -> ps))
3		19.22 722	. . 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 131	. . . 4 |- ((ph <-> ps) -> (ps -> ph))
6	5	19.20i 691	. . 3 |- (A.x(ph <-> ps) -> A.x(ps -> ph))
7		19.22 722	. . 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 397	1 |- (A.x(ph <-> ps) -> (E.xph <-> E.xps))

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

This theorem is referenced by:  biex 733  19.19 737  biexd 783

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-gen 677

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

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