Theorem ax12 1935

Description: Derive ax-12 1925 from ax12v 1926 via ax12o 1934. This shows that the weakening in ax12v 1926 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.)

Assertion

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

Proof of Theorem ax12

Step	Hyp	Ref	Expression
1		sp 1747	. . . . . 6 |- (A.x x = y -> x = y)
2	1	con3i 127	. . . . 5 |- (-. x = y -> -. A.x x = y)
3	2	adantr 451	. . . 4 |- ((-. x = y /\ y = z) -> -. A.x x = y)
4		equtrr 1683	. . . . . . . 8 |- (z = y -> (x = z -> x = y))
5	4	equcoms 1681	. . . . . . 7 |- (y = z -> (x = z -> x = y))
6	5	con3rr3 128	. . . . . 6 |- (-. x = y -> (y = z -> -. x = z))
7	6	imp 418	. . . . 5 |- ((-. x = y /\ y = z) -> -. x = z)
8		sp 1747	. . . . 5 |- (A.x x = z -> x = z)
9	7, 8	nsyl 113	. . . 4 |- ((-. x = y /\ y = z) -> -. A.x x = z)
10		ax12o 1934	. . . 4 |- (-. A.x x = y -> (-. A.x x = z -> (y = z -> A.x y = z)))
11	3, 9, 10	sylc 56	. . 3 |- ((-. x = y /\ y = z) -> (y = z -> A.x y = z))
12	11	ex 423	. 2 |- (-. x = y -> (y = z -> (y = z -> A.x y = z)))
13	12	pm2.43d 44	1 |- (-. x = y -> (y = z -> A.x y = z))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)