Theorem ax11v 1820

Description: This is a version of ax-11o 1816 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)

Assertion

Ref	Expression
ax11v	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax11v

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1689	. 2 |- E. z z = y
2		ax-17 1519	. . . . 5 |- ( ph -> A. z ph )
3		ax-11 1499	. . . . 5 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
4	2, 3	syl5 32	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
5		equequ2 1706	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
6	5	imbi1d 230	. . . . . . 7 |- ( z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
7	6	albidv 1817	. . . . . 6 |- ( z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
8	7	imbi2d 229	. . . . 5 |- ( z = y -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
9	5, 8	imbi12d 233	. . . 4 |- ( z = y -> ( ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) <-> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
10	4, 9	mpbii 147	. . 3 |- ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
11	10	exlimiv 1591	. 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 1346   = wceq 1348   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-17 1519  ax-i9 1523

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  equs5or  1823  sb56  1878