Theorem sb8 2134

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 2064	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1948	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 1981	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 184   A.wal 1368   F/wnf 1590   [wsb 1702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703

This theorem is referenced by:  sbhb  2152  sbnf2  2153  sbnf2OLD  2154  sb8eu  2301  sb8euOLD  2302  sb8euOLDOLD  2303  sb8iota  5497  mo5f  26021  wl-sb8eut  28547  sbcalf  29069  ax11-pm2  32677  bj-nfcf  32760