Theorem ax16ALT 1780

Description: Version of ax16 1734 that doesn't require ax-10 1436 or ax-12 1442 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 1694	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-17 1459	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 1729	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	ax16i 1779	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-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  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:  dvelimALT  1927  dvelimfv  1928