Metamath Proof Explorer

Theorem equidq

Description: equid with universal quantifier without using ax-c5 or ax-5 . (Contributed by NM, 13-Jan-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion equidq 
|- A. y x = x

Proof

Step Hyp Ref Expression
1 equidqe 
 |-  -. A. y -. x = x
2 ax10fromc7 
 |-  ( -. A. y x = x -> A. y -. A. y x = x )
3 hbequid 
 |-  ( x = x -> A. y x = x )
4 3 con3i 
 |-  ( -. A. y x = x -> -. x = x )
5 2 4 alrimih 
 |-  ( -. A. y x = x -> A. y -. x = x )
6 1 5 mt3 
 |-  A. y x = x