Theorem axc16ALT 2206

Description: Alternate proof of axc16 2035, shorter but requiring ax-11 1931 and using df-sb 1809. (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 2094	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-5 1769	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 2204	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	axc16i 2167	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1453   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809

This theorem is referenced by:  axc16gALT  2207