Theorem ax12OLD 2341

Description: Obsolete proof of ax12 2304 as of 4-Jul-2021 . Rederivation of axiom ax-12 2047 from ax12v 2048, axc11r 2187, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax12OLD	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Proof of Theorem ax12OLD

Step	Hyp	Ref	Expression
1		biidd 252	. . . . 5 |- ( A. x x = y -> ( ph <-> ph ) )
2	1	dral1 2325	. . . 4 |- ( A. x x = y -> ( A. x ph <-> A. y ph ) )
3		ax-1 6	. . . . 5 |- ( ph -> ( x = y -> ph ) )
4	3	alimi 1739	. . . 4 |- ( A. x ph -> A. x ( x = y -> ph ) )
5	2, 4	syl6bir 244	. . 3 |- ( A. x x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
6	5	a1d 25	. 2 |- ( A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
7		sp 2053	. . 3 |- ( A. y ph -> ph )
8		axc15 2303	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
9	7, 8	syl7 74	. 2 |- ( -. A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
10	6, 9	pm2.61i 176	1 |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> 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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)