Theorem cbv3 1730

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1732 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)

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 1564	. 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 1726	. 2 |- ( A. x ph -> ps )
6	2, 5	alrimi 1510	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  cbv3h  1731  cbv3v  1732  cbv1  1733  mo2n  2042  mo23  2055  setindis  13859  bdsetindis  13861