Theorem ax11vALT 2097

Description: Alternate proof of ax11v 2096 that avoids theorem ax16 2045 and is proved directly from ax-11 1746 rather than via ax11o 1994. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax11vALT	|- (x = y -> (ph -> A.x(x = y -> ph)))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem ax11vALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1951	. 2 |- E.z z = y
2		ax-17 1616	. . . . 5 |- (ph -> A.zph)
3		ax-11 1746	. . . . 5 |- (x = z -> (A.zph -> A.x(x = z -> ph)))
4	2, 3	syl5 28	. . . 4 |- (x = z -> (ph -> A.x(x = z -> ph)))
5		equequ2 1686	. . . . 5 |- (z = y -> (x = z <-> x = y))
6	5	imbi1d 308	. . . . . . 7 |- (z = y -> ((x = z -> ph) <-> (x = y -> ph)))
7	6	albidv 1625	. . . . . 6 |- (z = y -> (A.x(x = z -> ph) <-> A.x(x = y -> ph)))
8	7	imbi2d 307	. . . . 5 |- (z = y -> ((ph -> A.x(x = z -> ph)) <-> (ph -> A.x(x = y -> ph))))
9	5, 8	imbi12d 311	. . . 4 |- (z = y -> ((x = z -> (ph -> A.x(x = z -> ph))) <-> (x = y -> (ph -> A.x(x = y -> ph)))))
10	4, 9	mpbii 202	. . 3 |- (z = y -> (x = y -> (ph -> A.x(x = y -> ph))))
11	10	exlimiv 1634	. 2 |- (E.z z = y -> (x = y -> (ph -> A.x(x = y -> ph))))
12	1, 11	ax-mp 5	1 |- (x = y -> (ph -> A.x(x = y -> ph)))

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541   = wceq 1642

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

This theorem is referenced by: (None)