Theorem sb8 1909

Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb8e.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		sb8e.1	. 2 |- F/ y ph
2	1	nfs1 1862	. 2 |- F/ x [ y / x ] ph
3		sbequ12 1824	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbval 1807	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 105   A.wal 1400   F/wnf 1513   [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbnf2  2041  sb8eu  2099  nfraldya  2585  rabeq0  3552  abeq0  3553  sb8iota  5340