Theorem 19.21t 1473

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		alim 1340	. . . . 5 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
2	1	imim2d 28	. . . 4 |- (A.x(ph -> ps) -> ((ph -> A.xph) -> (ph -> A.xps)))
3	2	com12 14	. . 3 |- ((ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
4	3	a4s 1330	. 2 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
5		hba1 1350	. . . 4 |- (A.x(ph -> A.xph) -> A.xA.x(ph -> A.xph))
6		ax-4 1319	. . . 4 |- (A.x(ph -> A.xph) -> (ph -> A.xph))
7		hba1 1350	. . . . 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 1468	. . 3 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> A.xps)))
10		ax-4 1319	. . . . 5 |- (A.xps -> ps)
11	10	imim2i 11	. . . 4 |- ((ph -> A.xps) -> (ph -> ps))
12	11	alimi 1338	. . 3 |- (A.x(ph -> A.xps) -> A.x(ph -> ps))
13	9, 12	syl6 25	. 2 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> ps)))
14	4, 13	impbid 574	1 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) <-> (ph -> A.xps)))

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

This theorem is referenced by:  sbcom 1632  sbal2 1749  ax11indalem 1759  ax11inda2ALT 1760  r19.21t 2177  sbciegft 2482

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  ax-6o 1324

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

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