Theorem equs45f 1730

Description: Two ways of expressing substitution when y is not free in ph. (Contributed by NM, 25-Apr-2008.)

Hypothesis

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

Assertion

Ref	Expression
equs45f	|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Proof of Theorem equs45f

Step	Hyp	Ref	Expression
1		equs45f.1	. . . . 5 |- ( ph -> A. y ph )
2	1	anim2i 334	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ A. y ph ) )
3	2	eximi 1536	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y ph ) )
4		equs5a 1722	. . 3 |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
5	3, 4	syl 14	. 2 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) )
6		equs4 1660	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7	5, 6	impbii 124	1 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   E.wex 1426

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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-11 1442  ax-4 1445  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  sb5f  1732