Theorem ax12inda2 34232

Description: Induction step for constructing a substitution instance of ax-c15 34174 without using ax-c15 34174. Quantification case. When z and y are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 34233. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12inda2.1	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Assertion

Ref	Expression
ax12inda2	|- ( -. A. x x = y -> ( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) )

Distinct variable group:   y, z

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

Proof of Theorem ax12inda2

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . 5 |- ( A. z ph -> ( x = y -> A. z ph ) )
2		axc16g-o 34219	. . . . 5 |- ( A. y y = z -> ( ( x = y -> A. z ph ) -> A. x ( x = y -> A. z ph ) ) )
3	1, 2	syl5 34	. . . 4 |- ( A. y y = z -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) )
4	3	a1d 25	. . 3 |- ( A. y y = z -> ( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) )
5	4	a1d 25	. 2 |- ( A. y y = z -> ( -. A. x x = y -> ( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) )
6		ax12inda2.1	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7	6	ax12indalem 34230	. 2 |- ( -. A. y y = z -> ( -. A. x x = y -> ( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) )
8	5, 7	pm2.61i 176	1 |- ( -. A. x x = y -> ( x = y -> ( A. z ph -> A. x ( x = y -> A. z 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-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175  ax-c16 34177

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  ax12inda  34233