Theorem ax12v2 2049

Description: It is possible to remove any restriction on ph in ax12v 2048. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2048 instead when sufficient. (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.)

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 1949	. . 3 |- ( y = z -> ( x = y -> x = z ) )
2		ax12v 2048	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
3	1	imim1d 82	. . . . 5 |- ( y = z -> ( ( x = z -> ph ) -> ( x = y -> ph ) ) )
4	3	alimdv 1845	. . . 4 |- ( y = z -> ( A. x ( x = z -> ph ) -> A. x ( x = y -> ph ) ) )
5	2, 4	syl9r 78	. . 3 |- ( y = z -> ( x = z -> ( ph -> A. x ( x = y -> ph ) ) ) )
6	1, 5	syld 47	. 2 |- ( y = z -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7		ax6evr 1942	. 2 |- E. z y = z
8	6, 7	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:  axc11rvOLD  2140  sb56  2150  bj-ax12  32634  wl-lem-exsb  33348  wl-lem-moexsb  33350