Theorem dvelimALT 2133

Description: Version of dvelim 2016 that doesn't use ax-10 2140. (See dvelimh 1964 for a version that doesn't use ax-11 1746.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ps,z   x,z   y,z

Allowed substitution hints:   ph(x,y,z)   ps(x,y)

Proof of Theorem dvelimALT

Step	Hyp	Ref	Expression
1		ax-17 1616	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		ax16ALT 2047	. . . . 5 |- (A.x x = z -> ((z = y -> ph) -> A.x(z = y -> ph)))
3	2	a1d 22	. . . 4 |- (A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
4		hbn1 1730	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
5		hbn1 1730	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
6	4, 5	hban 1828	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
7		ax12o 1934	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
8	7	imp 418	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
9		dvelimALT.1	. . . . . . 7 |- (ph -> A.xph)
10	9	a1i 10	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (ph -> A.xph))
11	6, 8, 10	hbimd 1815	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
12	11	ex 423	. . . 4 |- (-. A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
13	3, 12	pm2.61i 156	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
14	1, 13	hbald 1740	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
15		ax-17 1616	. . 3 |- (ps -> A.zps)
16		dvelimALT.2	. . 3 |- (z = y -> (ph <-> ps))
17	15, 16	equsalh 1961	. 2 |- (A.z(z = y -> ph) <-> ps)
18	17	albii 1566	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
19	14, 17, 18	3imtr3g 260	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  dveeq2-o16  2185