Theorem cbv3 1764

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 1766 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 1598	. 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 1760	. 2 |- ( A. x ph -> ps )
6	2, 5	alrimi 1544	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1370   F/wnf 1482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-i9 1552  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  cbv3h  1765  cbv3v  1766  cbv1  1767  mo2n  2081  mo23  2094  setindis  15836  bdsetindis  15838