Theorem ax16 1734

Description: Theorem showing that ax-16 1735 is redundant if ax-17 1459 is included in the axiom system. The important part of the proof is provided by aev 1733.

See ax16ALT 1780 for an alternate proof that does not require ax-10 1436 or ax-12 1442.

This theorem should not be referenced in any proof. Instead, use ax-16 1735 below so that theorems needing ax-16 1735 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 1733	. 2 |- ( A. x x = y -> A. z x = z )
2		ax-17 1459	. . . 4 |- ( ph -> A. z ph )
3		sbequ12 1694	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	3	biimpcd 157	. . . 4 |- ( ph -> ( x = z -> [ z / x ] ph ) )
5	2, 4	alimdh 1396	. . 3 |- ( ph -> ( A. z x = z -> A. z [ z / x ] ph ) )
6	2	hbsb3 1729	. . . 4 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
7		stdpc7 1693	. . . 4 |- ( z = x -> ( [ z / x ] ph -> ph ) )
8	6, 2, 7	cbv3h 1671	. . 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 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  dveeq2  1736  dveeq2or  1737  a16g  1785  exists2  2038