Theorem sb8 2151

Description: Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/ y ph

Assertion

Ref	Expression
sb8	|- ( A. x ph <-> A. y [ y / x ] ph )

Proof of Theorem sb8

Step	Hyp	Ref	Expression
1		sb5rf.1	. 2 |- F/ y ph
2	1	nfs1 2088	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1976	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 2005	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 184   A.wal 1379   F/wnf 1601   [wsb 1724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1598  df-nf 1602  df-sb 1725

This theorem is referenced by:  sbhb  2166  sbnf2  2167  sb8eu  2302  sb8euOLD  2303  sb8iota  5544  mo5f  27248  wl-sb8eut  29990  sbcalf  30485  ax11-pm2  34121  bj-nfcf  34204