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Metamath Proof Explorer

Theorem equs4

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition ( sbalex ) or a nonfreeness hypothesis ( equs45f ). Usage of this theorem is discouraged because it depends on ax-13 . See equs4v for a weaker version requiring fewer axioms. (Contributed by NM, 10-May-1993) (Proof shortened by Mario Carneiro, 20-May-2014) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

Ref Expression
Assertion equs4 
|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )

Proof

Step Hyp Ref Expression
1 ax6e 
 |-  E. x x = y
2 exintr 
 |-  ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ( x = y /\ ph ) ) )
3 1 2 mpi 
 |-  ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )