Theorem equs4 2063

Description: Lemma used in proofs of substitution properties. The converse requires either a dv condition (sb56 2198) or a non-freeness hypothesis (equs45f 2117). See equs4v 1813 for a 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.)

Assertion

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

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		ax6e 2031	. 2 |- E. x x = y
2		exintr 1725	. 2 |- ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ( x = y /\ ph ) ) )
3	1, 2	mpi 21	1 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1405   E.wex 1635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-12 1880  ax-13 2028

This theorem depends on definitions:  df-bi 187  df-an 371  df-ex 1636

This theorem is referenced by:  equsex  2066  equs45f  2117  equs5  2118  sb2  2119