Theorem ax16ALT 1907

Description: Version of ax16 1861 that does not require ax-10 1553 or ax12 1560 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 1819	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-17 1574	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 1856	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	ax16i 1906	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1395   [wsb 1810

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  dvelimALT  2063  dvelimfv  2064