Theorem 19.21g 792

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

Assertion

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

Proof of Theorem 19.21g

Step	Hyp	Ref	Expression
1		19.20 690	. . . . 5 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
2	1	syl3d 26	. . . 4 |- (A.x(ph -> ps) -> ((ph -> A.xph) -> (ph -> A.xps)))
3	2	com12 13	. . 3 |- ((ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
4	3	a4s 682	. 2 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) -> (ph -> A.xps)))
5		hba1 698	. . . 4 |- (A.x(ph -> A.xph) -> A.xA.x(ph -> A.xph))
6		ax-4 673	. . . 4 |- (A.x(ph -> A.xph) -> (ph -> A.xph))
7		hba1 698	. . . . 5 |- (A.xps -> A.xA.xps)
8	7	a1i 7	. . . 4 |- (A.x(ph -> A.xph) -> (A.xps -> A.xA.xps))
9	5, 6, 8	hbimd 787	. . 3 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> A.xps)))
10		ax-4 673	. . . . 5 |- (A.xps -> ps)
11	10	syl3 18	. . . 4 |- ((ph -> A.xps) -> (ph -> ps))
12	11	19.20i 691	. . 3 |- (A.x(ph -> A.xps) -> A.x(ph -> ps))
13	9, 12	syl6 23	. 2 |- (A.x(ph -> A.xph) -> ((ph -> A.xps) -> A.x(ph -> ps)))
14	4, 13	impbid 397	1 |- (A.x(ph -> A.xph) -> (A.x(ph -> ps) <-> (ph -> A.xps)))

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

This theorem is referenced by:  sbcom 916  sbal2 1005

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-6 675  ax-gen 677

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

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