Theorem equs4 2290

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a dv condition (sb56 2150) or a non-freeness hypothesis (equs45f 2350). See equs4v 1930 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 2250	. 2 |- E. x x = y
2		exintr 1819	. 2 |- ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ( x = y /\ ph ) ) )
3	1, 2	mpi 20	1 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  equsex  2292  equs45f  2350  equs5  2351  sb2  2352  bj-sbsb  32824