Metamath Proof Explorer


Theorem equs45f

Description: Two ways of expressing substitution when y is not free in ph . The implication "to the left" is equs4 and does not require the nonfreeness hypothesis. Theorem sbalex replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 replaces it with a distinctor as antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 25-Apr-2008) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis equs45f.1
|- F/ y ph
Assertion equs45f
|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Proof

Step Hyp Ref Expression
1 equs45f.1
 |-  F/ y ph
2 1 nf5ri
 |-  ( ph -> A. y ph )
3 2 anim2i
 |-  ( ( x = y /\ ph ) -> ( x = y /\ A. y ph ) )
4 3 eximi
 |-  ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y ph ) )
5 equs5a
 |-  ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
6 4 5 syl
 |-  ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) )
7 equs4
 |-  ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
8 6 7 impbii
 |-  ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )