Theorem ax12 2304

Description: Rederivation of axiom ax-12 2047 from ax12v 2048 (used only via sp 2053) , axc11r 2187, and axc15 2303 (on top of Tarski's FOL). (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.)

Assertion

Ref	Expression
ax12	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Proof of Theorem ax12

Step	Hyp	Ref	Expression
1		axc11r 2187	. . . 4 |- ( A. x x = y -> ( A. y ph -> A. x ph ) )
2		ala1 1741	. . . 4 |- ( A. x ph -> A. x ( x = y -> ph ) )
3	1, 2	syl6 35	. . 3 |- ( A. x x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
4	3	a1d 25	. 2 |- ( A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
5		sp 2053	. . 3 |- ( A. y ph -> ph )
6		axc15 2303	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7	5, 6	syl7 74	. 2 |- ( -. A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
8	4, 7	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:  equs5a  2348  equs5e  2349  bj-ax12v3  32675  axc11-o  34236