Theorem ddelimf2 907

Description: Proof of ddelimf 908 without using ax-11 801. This may be useful in a study to determine whether ax-11 801 can be derived from the others, which is currently unknown.

Hypotheses

Ref	Expression
ddelimf2.1	|- (ph -> A.xph)
ddelimf2.2	|- (ps -> A.zps)
ddelimf2.3	|- (z = y -> (ph <-> ps))

Assertion

Ref	Expression
ddelimf2	|- (-. A.x x = y -> (ps -> A.xps))

Proof of Theorem ddelimf2

Step	Hyp	Ref	Expression
1		ax-10 800	. . . . . 6 |- (A.z z = x -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
2	1	eq4s 822	. . . . 5 |- (A.x x = z -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
3		hba1 698	. . . . 5 |- (A.z(z = y -> ph) -> A.zA.z(z = y -> ph))
4	2, 3	syl5 22	. . . 4 |- (A.x x = z -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
5	4	a1d 14	. . 3 |- (A.x x = z -> (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph))))
6		eq6 826	. . . . . 6 |- (-. A.x x = z -> A.z -. A.x x = z)
7		eq6 826	. . . . . 6 |- (-. A.x x = y -> A.z -. A.x x = y)
8	6, 7	hban 704	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> A.z(-. A.x x = z /\ -. A.x x = y))
9		eq6 826	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
10		eq6 826	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	9, 10	hban 704	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax-12 802	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 277	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
14		ddelimf2.1	. . . . . . 7 |- (ph -> A.xph)
15	14	a1i 7	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (ph -> A.xph))
16	11, 13, 15	hbimd 787	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
17	8, 16	hbald 790	. . . 4 |- ((-. A.x x = z /\ -. A.x x = y) -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
18	17	exp 291	. . 3 |- (-. A.x x = z -> (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph))))
19	5, 18	pm2.61i 110	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
20		ddelimf2.2	. . 3 |- (ps -> A.zps)
21		ddelimf2.3	. . 3 |- (z = y -> (ph <-> ps))
22	20, 21	eqsal 833	. 2 |- (A.z(z = y -> ph) <-> ps)
23	22	bial 695	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
24	19, 22, 23	3imtr3g 425	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802

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

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