Theorem cbv3 1630

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
cbv3	|- ( A. x ph -> A. y ps )

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . 3 |- F/ y ph
2	1	nfal 1468	. 2 |- F/ y A. x ph
3		cbv3.2	. . 3 |- F/ x ps
4		cbv3.3	. . 3 |- ( x = y -> ( ph -> ps ) )
5	3, 4	spim 1626	. 2 |- ( A. x ph -> ps )
6	2, 5	alrimi 1415	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbv3h  1631  cbv1  1632  mo2n  1928  mo23  1941  setindis  10092  bdsetindis  10094