Metamath Proof Explorer


Theorem bj-equsalhv

Description: Version of equsalh with a disjoint variable condition, which does not require ax-13 . Remark: this is the same as equsalhw . TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw has been moved to Main; Theorem ax13lem2 has a DV version which is a simple consequence of ax5e ; Theorems nfeqf2 , dveeq2 , nfeqf1 , dveeq1 , nfeqf , axc9 , ax13 , have dv versions which are simple consequences of ax-5 . (Contributed by BJ, 14-Jun-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses bj-equsalhv.nf
|- ( ps -> A. x ps )
bj-equsalhv.1
|- ( x = y -> ( ph <-> ps ) )
Assertion bj-equsalhv
|- ( A. x ( x = y -> ph ) <-> ps )

Proof

Step Hyp Ref Expression
1 bj-equsalhv.nf
 |-  ( ps -> A. x ps )
2 bj-equsalhv.1
 |-  ( x = y -> ( ph <-> ps ) )
3 1 nf5i
 |-  F/ x ps
4 3 2 equsalv
 |-  ( A. x ( x = y -> ph ) <-> ps )