Theorem axc16ALT 2109

Description: Alternate proof of axc16 1949, shorter but requiring ax-11 1850 and using df-sb 1748. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16ALT	|- ( A. x x = y -> ( ph -> A. x ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem axc16ALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbequ12 2000	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-5 1712	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 2107	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	axc16i 2070	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1397   [wsb 1747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748

This theorem is referenced by:  axc16gALT  2110