Theorem ax16ALT 1832

Description: Version of ax16 1786 that doesn't require ax-10 1484 or ax-12 1490 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax16ALT	|- ( A. x x = y -> ( ph -> A. x ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax16ALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbequ12 1745	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-17 1507	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 1781	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	ax16i 1831	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1330   [wsb 1736

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  dvelimALT  1986  dvelimfv  1987