Theorem ax16ALT 1870

Description: Version of ax16 1824 that does not require ax-10 1516 or ax12 1523 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 1782	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-17 1537	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 1819	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	ax16i 1869	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1362   [wsb 1773

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  dvelimALT  2022  dvelimfv  2023