Theorem sb6f 1775

Description: Equivalence for substitution when y is not free in ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)

Hypothesis

Ref	Expression
equs45f.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
sb6f	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Proof of Theorem sb6f

Step	Hyp	Ref	Expression
1		equs45f.1	. . . 4 |- ( ph -> A. y ph )
2	1	sbimi 1737	. . 3 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		sb4a 1773	. . 3 |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
4	2, 3	syl 14	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> ph ) )
5		sb2 1740	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
6	4, 5	impbii 125	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-11 1484  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sb5f  1776  sbcof2  1782