Theorem sb8h 1878

Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb8h.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
sb8h	|- ( A. x ph <-> A. y [ y / x ] ph )

Proof of Theorem sb8h

Step	Hyp	Ref	Expression
1		sb8h.1	. 2 |- ( ph -> A. y ph )
2	1	hbsb3 1832	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		sbequ12 1795	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1, 2, 3	cbvalh 1777	1 |- ( A. x ph <-> A. y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   [wsb 1786

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbhb  1969  sb8euh  2078