Theorem ax12vOLDOLD 2051

Description: Obsolete proof of ax12v 2048 as of 7-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.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax12vOLDOLD	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax12vOLDOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1953	. . . 4 |- ( z = y -> ( x = z <-> x = y ) )
2	1	biimprd 238	. . 3 |- ( z = y -> ( x = y -> x = z ) )
3		ax-5 1839	. . . . 5 |- ( ph -> A. z ph )
4		ax-12 2047	. . . . 5 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
5	3, 4	syl5 34	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
6	2	imim1d 82	. . . . 5 |- ( z = y -> ( ( x = z -> ph ) -> ( x = y -> ph ) ) )
7	6	alimdv 1845	. . . 4 |- ( z = y -> ( A. x ( x = z -> ph ) -> A. x ( x = y -> ph ) ) )
8	5, 7	syl9r 78	. . 3 |- ( z = y -> ( x = z -> ( ph -> A. x ( x = y -> ph ) ) ) )
9	2, 8	syld 47	. 2 |- ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
10		ax6ev 1890	. 2 |- E. z z = y
11	9, 10	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)