Theorem cbv3h 1983

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
cbv3h.1	|- (ph -> A.yph)
cbv3h.2	|- (ps -> A.xps)
cbv3h.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbv3h	|- (A.xph -> A.yps)

Proof of Theorem cbv3h

Step	Hyp	Ref	Expression
1		cbv3h.1	. . . 4 |- (ph -> A.yph)
2	1	a1i 10	. . 3 |- (y = y -> (ph -> A.yph))
3		cbv3h.2	. . . 4 |- (ps -> A.xps)
4	3	a1i 10	. . 3 |- (y = y -> (ps -> A.xps))
5		cbv3h.3	. . . 4 |- (x = y -> (ph -> ps))
6	5	a1i 10	. . 3 |- (y = y -> (x = y -> (ph -> ps)))
7	2, 4, 6	cbv1h 1978	. 2 |- (A.xA.y y = y -> (A.xph -> A.yps))
8		stdpc6 1687	. 2 |- A.y y = y
9	7, 8	mpg 1548	1 |- (A.xph -> A.yps)

Syntax hints:   -> 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

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:  cleqh  2450