Theorem sb8 2253

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 2194	. 2 |- F/ x [ y / x ] ph
3		sbequ12 2083	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 2114	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 188   A.wal 1442   F/wnf 1667   [wsb 1797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798

This theorem is referenced by:  sbhb  2267  sbnf2  2268  sb8eu  2332  sb8iota  5553  mo5f  28120  ax11-pm2  31438  bj-nfcf  31527  wl-sb8eut  31906  sbcalf  32352