Theorem axc16ALT 2366

Description: Alternate proof of axc16 2135, shorter but requiring ax-10 2019, ax-11 2034, ax-13 2246 and using df-nf 1710 and df-sb 1881. (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 2111	. 2 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
2		ax-5 1839	. . 3 |- ( ph -> A. z ph )
3	2	hbsb3 2364	. 2 |- ( [ z / x ] ph -> A. x [ z / x ] ph )
4	1, 3	axc16i 2322	1 |- ( A. x x = y -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  axc16gALT  2367