Theorem ax16ALT 1782

Description: Version of ax16 1736 that doesn't require ax-10 1437 or ax-12 1443 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 1696	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-17 1460	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 1731	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	ax16i 1781	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1283   [wsb 1687

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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688

This theorem is referenced by:  dvelimALT  1929  dvelimfv  1930