Theorem cbv3 1982

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2142. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbv3.1	|- F/yph
cbv3.2	|- F/xps
cbv3.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbv3	|- (A.xph -> A.yps)

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . . 4 |- F/yph
2	1	a1i 10	. . 3 |- ( T. -> F/yph)
3		cbv3.2	. . . 4 |- F/xps
4	3	a1i 10	. . 3 |- ( T. -> F/xps)
5		cbv3.3	. . . 4 |- (x = y -> (ph -> ps))
6	5	a1i 10	. . 3 |- ( T. -> (x = y -> (ph -> ps)))
7	2, 4, 6	cbv1 1979	. 2 |- (A.xA.y T. -> (A.xph -> A.yps))
8		tru 1321	. . 3 |- T.
9	8	ax-gen 1546	. 2 |- A.y T.
10	7, 9	mpg 1548	1 |- (A.xph -> A.yps)

Syntax hints:   -> wi 4   T. wtru 1316  A.wal 1540   F/wnf 1544

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-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbval  1984  ax16i  2046  ax16ALT2  2048  sb8  2092  mo  2226