Theorem ax12inda2 2277

Description: Induction step for constructing a substitution instance of ax-c15 2220 without using ax-c15 2220. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 2278. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12inda2.1

Assertion

Ref	Expression
ax12inda2

Distinct variable group:   ,

Proof of Theorem ax12inda2

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . 5
2		ax16g-o 2264	. . . . 5
3	1, 2	syl5 32	. . . 4
4	3	a1d 25	. . 3
5	4	a1d 25	. 2
6		ax12inda2.1	. . 3
7	6	ax12indalem 2275	. 2
8	5, 7	pm2.61i 164	1

Syntax hints:  -.wn 3  ->wi 4  A.wal 1393

This theorem is referenced by:  ax12inda  2278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c11 2218  ax-c9 2221  ax-c16 2223

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617