Theorem cbv3h 1721

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 shortened by Wolf Lammen, 12-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
cbv3h	|- ( A. x ph -> A. y ps )

Proof of Theorem cbv3h

Step	Hyp	Ref	Expression
1		cbv3h.1	. . 3 |- ( ph -> A. y ph )
2	1	nfi 1438	. 2 |- F/ y ph
3		cbv3h.2	. . 3 |- ( ps -> A. x ps )
4	3	nfi 1438	. 2 |- F/ x ps
5		cbv3h.3	. 2 |- ( x = y -> ( ph -> ps ) )
6	2, 4, 5	cbv3 1720	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  cbvalh  1726  ax16  1785  ax16i  1830  cleqh  2237