Theorem ax11inda2 2199

Description: Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. When z and y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2200. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11inda2.1	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Assertion

Ref	Expression
ax11inda2	|- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Distinct variable group:   y,z

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem ax11inda2

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . 5 |- (A.zph -> (x = y -> A.zph))
2		a16g-o 2186	. . . . 5 |- (A.y y = z -> ((x = y -> A.zph) -> A.x(x = y -> A.zph)))
3	1, 2	syl5 28	. . . 4 |- (A.y y = z -> (A.zph -> A.x(x = y -> A.zph)))
4	3	a1d 22	. . 3 |- (A.y y = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
5	4	a1d 22	. 2 |- (A.y y = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
6		ax11inda2.1	. . 3 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
7	6	ax11indalem 2197	. 2 |- (-. A.y y = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
8	5, 7	pm2.61i 156	1 |- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

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  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax11inda  2200