Theorem sb8 2169

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 2106	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1997	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 2026	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 184   A.wal 1396   F/wnf 1621   [wsb 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745

This theorem is referenced by:  sbhb  2184  sbnf2  2185  sb8eu  2319  sb8euOLD  2320  sb8iota  5541  mo5f  27584  wl-sb8eut  30265  sbcalf  30760  ax11-pm2  34829  bj-nfcf  34912