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Theorem equs4 2092
Description: Lemma used in proofs of implicit substitution properties. The converse requires either a dv condition (sb56 2227) or a non-freeness hypothesis (equs45f 2148). See equs4v 1839 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
StepHypRef Expression
1 ax6e 2060 . 2  |-  E. x  x  =  y
2 exintr 1750 . 2  |-  ( A. x ( x  =  y  ->  ph )  -> 
( E. x  x  =  y  ->  E. x
( x  =  y  /\  ph ) ) )
31, 2mpi 20 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658
This theorem is referenced by:  equsex  2095  equs45f  2148  equs5  2149  sb2  2150
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