Theorem ax12vOLD 2050

Description: Obsolete proof of ax12v2 2049 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax12vOLD	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax12vOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equtrr 1949	. . 3 |- ( y = z -> ( x = y -> x = z ) )
2		ax-5 1839	. . . . 5 |- ( ph -> A. z ph )
3		ax-12 2047	. . . . 5 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
4	2, 3	syl5 34	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
5	1	imim1d 82	. . . . 5 |- ( y = z -> ( ( x = z -> ph ) -> ( x = y -> ph ) ) )
6	5	alimdv 1845	. . . 4 |- ( y = z -> ( A. x ( x = z -> ph ) -> A. x ( x = y -> ph ) ) )
7	4, 6	syl9r 78	. . 3 |- ( y = z -> ( x = z -> ( ph -> A. x ( x = y -> ph ) ) ) )
8	1, 7	syld 47	. 2 |- ( y = z -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
9		ax6evr 1942	. 2 |- E. z y = z
10	8, 9	exlimiiv 1859	1 |- ( x = y -> ( ph -> A. x ( x = y -> 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)