Theorem sb6f 1684

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 1647	. . 3 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		sb4a 1682	. . 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 1650	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
6	4, 5	impbii 117	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sb5f  1685  sbcof2  1691