Theorem axc16gOLD 2161

Description: Obsolete proof of axc16g 2134 as of 11-Oct-2021. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2139. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc16gOLD	|- ( A. x x = y -> ( ph -> A. z ph ) )

Distinct variable group:   x, y

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

Proof of Theorem axc16gOLD

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 1981	. 2 |- ( A. x x = y -> A. z z = w )
2		ax-5 1839	. 2 |- ( ph -> A. w ph )
3		axc11rvOLD 2140	. 2 |- ( A. z z = w -> ( A. w ph -> A. z ph ) )
4	1, 2, 3	syl2im 40	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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)