Theorem ax16 1859

Description: Theorem showing that ax-16 1860 is redundant if ax-17 1572 is included in the axiom system. The important part of the proof is provided by aev 1858.

See ax16ALT 1905 for an alternate proof that does not require ax-10 1551 or ax12 1558.

This theorem should not be referenced in any proof. Instead, use ax-16 1860 below so that theorems needing ax-16 1860 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Assertion

Ref	Expression
ax16	|- ( A. x x = y -> ( ph -> A. x ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax16

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1858	. 2 |- ( A. x x = y -> A. z x = z )
2		ax-17 1572	. . . 4 |- ( ph -> A. z ph )
3		sbequ12 1817	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	3	biimpcd 159	. . . 4 |- ( ph -> ( x = z -> [ z / x ] ph ) )
5	2, 4	alimdh 1513	. . 3 |- ( ph -> ( A. z x = z -> A. z [ z / x ] ph ) )
6	2	hbsb3 1854	. . . 4 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
7		stdpc7 1816	. . . 4 |- ( z = x -> ( [ z / x ] ph -> ph ) )
8	6, 2, 7	cbv3h 1789	. . 3 |- ( A. z [ z / x ] ph -> A. x ph )
9	5, 8	syl6com 35	. 2 |- ( A. z x = z -> ( ph -> A. x ph ) )
10	1, 9	syl 14	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1393   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  dveeq2  1861  dveeq2or  1862  a16g  1910  exists2  2175