Theorem ax12v2 1952

Description: It is possible to remove any restriction on ph in ax12v 1951. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1932 and ax-13 2104. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)

Assertion

Ref	Expression
ax12v2	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax12v2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equtrr 1874	. . 3 |- ( y = z -> ( x = y -> x = z ) )
2		ax12v 1951	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
3	1	imim1d 77	. . . . 5 |- ( y = z -> ( ( x = z -> ph ) -> ( x = y -> ph ) ) )
4	3	alimdv 1771	. . . 4 |- ( y = z -> ( A. x ( x = z -> ph ) -> A. x ( x = y -> ph ) ) )
5	2, 4	syl9r 73	. . 3 |- ( y = z -> ( x = z -> ( ph -> A. x ( x = y -> ph ) ) ) )
6	1, 5	syld 44	. 2 |- ( y = z -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7		ax6evr 1867	. 2 |- E. z y = z
8	6, 7	exlimiiv 1785	1 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672

This theorem is referenced by:  sb56  2096  bj-ax12  31311  wl-lem-exsb  31965  wl-lem-moexsb  31967