Theorem ax16 1694

Description: Theorem showing that ax-16 1695 is redundant if ax-17 1419 is included in the axiom system. The important part of the proof is provided by aev 1693.

See ax16ALT 1739 for an alternate proof that does not require ax-10 1396 or ax-12 1402.

This theorem should not be referenced in any proof. Instead, use ax-16 1695 below so that theorems needing ax-16 1695 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 1693	. 2 |- ( A. x x = y -> A. z x = z )
2		ax-17 1419	. . . 4 |- ( ph -> A. z ph )
3		sbequ12 1654	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	3	biimpcd 148	. . . 4 |- ( ph -> ( x = z -> [ z / x ] ph ) )
5	2, 4	alimdh 1356	. . 3 |- ( ph -> ( A. z x = z -> A. z [ z / x ] ph ) )
6	2	hbsb3 1689	. . . 4 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
7		stdpc7 1653	. . . 4 |- ( z = x -> ( [ z / x ] ph -> ph ) )
8	6, 2, 7	cbv3h 1631	. . 3 |- ( A. z [ z / x ] ph -> A. x ph )
9	5, 8	syl6com 31	. 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 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  dveeq2  1696  dveeq2or  1697  a16g  1744  exists2  1997