Theorem 19.21t 1115

Description: Closed form of Theorem 19.21 of [Margaris] p. 90.

Assertion

Ref	Expression
19.21t	|- (A.x(ph -> A.xph) -> (A.x(ph -> ps) <-> (ph -> A.xps)))

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		19.20 994	. . . . 5 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
2	1	imim2d 25	. . . 4 |- (A.x(ph -> ps) -> ((ph -> A.xph) -> (ph -> A.xps)))
3	2	com12 11	. . 3 |- ((ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
4	3	a4s 984	. 2 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
5		hba1 1003	. . . 4 |- (A.x(ph -> A.xph) -> A.xA.x(ph -> A.xph))
6		ax-4 973	. . . 4 |- (A.x(ph -> A.xph) -> (ph -> A.xph))
7		hba1 1003	. . . . 5 |- (A.xps -> A.xA.xps)
8	7	a1i 8	. . . 4 |- (A.x(ph -> A.xph) -> (A.xps -> A.xA.xps))
9	5, 6, 8	hbimd 1110	. . 3 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> A.xps)))
10		ax-4 973	. . . . 5 |- (A.xps -> ps)
11	10	imim2i 17	. . . 4 |- ((ph -> A.xps) -> (ph -> ps))
12	11	19.20i 992	. . 3 |- (A.x(ph -> A.xps) -> A.x(ph -> ps))
13	9, 12	syl6 22	. 2 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> ps)))
14	4, 13	impbid 516	1 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) <-> (ph -> A.xps)))

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

This theorem is referenced by:  sbcom 1258  sbal2 1358  ax11indalem 1368  ax11inda2ALT 1369  r19.21t 1715  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

Copyright terms: Public domain