Theorem axc16gALT 2367

Description: Alternate proof of axc16g 2134 that uses df-sb 1881 and requires ax-10 2019, ax-11 2034, ax-13 2246. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16gALT	|- ( A. x x = y -> ( ph -> A. z ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem axc16gALT

Step	Hyp	Ref	Expression
1		aev 1983	. 2 |- ( A. x x = y -> A. z z = x )
2		axc16ALT 2366	. 2 |- ( A. x x = y -> ( ph -> A. x ph ) )
3		biidd 252	. . . 4 |- ( A. z z = x -> ( ph <-> ph ) )
4	3	dral1 2325	. . 3 |- ( A. z z = x -> ( A. z ph <-> A. x ph ) )
5	4	biimprd 238	. 2 |- ( A. z z = x -> ( A. x ph -> A. z ph ) )
6	1, 2, 5	sylsyld 61	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

Syntax hints:   -> wi 4   A.wal 1481

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: (None)