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Metamath Proof Explorer

Theorem equs5a

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . This proof uses ax12 , see equs5aALT for an alternative one using ax-12 but not ax13 . Usage of the weaker equs5av is preferred, which uses ax12v2 , but not ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

Ref Expression
Assertion equs5a 
|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Proof

Step Hyp Ref Expression
1 nfa1 
 |-  F/ x A. x ( x = y -> ph )
2 ax12 
 |-  ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3 2 imp 
 |-  ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
4 1 3 exlimi 
 |-  ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )