Theorem sb8 2218

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 2157	. 2 |- F/ x [ y / x ] ph
3		sbequ12 2047	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 2075	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 187   A.wal 1435   F/wnf 1663   [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787

This theorem is referenced by:  sbhb  2233  sbnf2  2234  sb8eu  2299  sb8iota  5569  mo5f  28106  ax11-pm2  31396  bj-nfcf  31484  wl-sb8eut  31820  sbcalf  32266