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Theorem mosub 3114
Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
Hypothesis
Ref Expression
mosub.1  |-  E* x ph
Assertion
Ref Expression
mosub  |-  E* x E. y ( y  =  A  /\  ph )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem mosub
StepHypRef Expression
1 moeq 3112 . 2  |-  E* y 
y  =  A
2 mosub.1 . . 3  |-  E* x ph
32ax-gen 1556 . 2  |-  A. y E* x ph
4 moexexv 2353 . 2  |-  ( ( E* y  y  =  A  /\  A. y E* x ph )  ->  E* x E. y ( y  =  A  /\  ph ) )
51, 3, 4mp2an 655 1  |-  E* x E. y ( y  =  A  /\  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653   E*wmo 2284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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