Theorem ax12olem6 1932

Description: Lemma for ax12o 1934. Derivation of ax12o 1934 from the hypotheses, without using ax12o 1934. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)

Hypotheses

Ref	Expression
ax12olem6.1	|- (-. A.x x = z -> (z = w -> A.x z = w))
ax12olem6.2	|- (-. A.x x = y -> (y = w -> A.x y = w))

Assertion

Ref	Expression
ax12olem6	|- (-. A.x x = y -> (-. A.x x = z -> (y = z -> A.x y = z)))

Distinct variable groups:   x,w   y,w   z,w

Proof of Theorem ax12olem6

Step	Hyp	Ref	Expression
1		hbn1 1730	. . . . 5 |- (-. A.x x = z -> A.x -. A.x x = z)
2		ax12olem6.1	. . . . 5 |- (-. A.x x = z -> (z = w -> A.x z = w))
3	1, 2	hbim1 1810	. . . 4 |- ((-. A.x x = z -> z = w) -> A.x(-. A.x x = z -> z = w))
4		ax-17 1616	. . . 4 |- ((-. A.x x = z -> y = z) -> A.w(-. A.x x = z -> y = z))
5		equcom 1680	. . . . . 6 |- (z = w <-> w = z)
6		equequ1 1684	. . . . . 6 |- (w = y -> (w = z <-> y = z))
7	5, 6	syl5bb 248	. . . . 5 |- (w = y -> (z = w <-> y = z))
8	7	imbi2d 307	. . . 4 |- (w = y -> ((-. A.x x = z -> z = w) <-> (-. A.x x = z -> y = z)))
9		ax12olem6.2	. . . 4 |- (-. A.x x = y -> (y = w -> A.x y = w))
10	3, 4, 8, 9	dvelimhw 1849	. . 3 |- (-. A.x x = y -> ((-. A.x x = z -> y = z) -> A.x(-. A.x x = z -> y = z)))
11	1	19.21h 1797	. . 3 |- (A.x(-. A.x x = z -> y = z) <-> (-. A.x x = z -> A.x y = z))
12	10, 11	syl6ib 217	. 2 |- (-. A.x x = y -> ((-. A.x x = z -> y = z) -> (-. A.x x = z -> A.x y = z)))
13	12	pm2.86d 93	1 |- (-. A.x x = y -> (-. A.x x = z -> (y = z -> A.x y = z)))

Syntax hints:  -. wn 3   -> wi 4  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

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

This theorem is referenced by:  ax12olem7  1933