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Metamath Proof Explorer

Theorem equs5

Description: Lemma used in proofs of substitution properties. If there is a disjoint variable condition on x , y , then sbalex can be used instead; if y is not free in ph , then equs45f can be used. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (Revised by BJ, 1-Oct-2018) (New usage is discouraged.)

Ref Expression
Assertion equs5 
|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )

Proof

Step Hyp Ref Expression
1 nfna1 
 |-  F/ x -. A. x x = y
2 nfa1 
 |-  F/ x A. x ( x = y -> ph )
3 axc15 
 |-  ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
4 3 impd 
 |-  ( -. A. x x = y -> ( ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
5 1 2 4 exlimd 
 |-  ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
6 equs4 
 |-  ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7 5 6 impbid1 
 |-  ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )