Theorem sb6f 1724

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 1687	. . 3 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		sb4a 1722	. . 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 1690	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
6	4, 5	impbii 124	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-11 1437  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sb5f  1725  sbcof2  1731