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Theorem mosub 2943
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 2941 . 2  |-  E* y 
y  =  A
2 mosub.1 . . 3  |-  E* x ph
32ax-gen 1533 . 2  |-  A. y E* x ph
4 moexexv 2213 . 2  |-  ( ( E* y  y  =  A  /\  A. y E* x ph )  ->  E* x E. y ( y  =  A  /\  ph ) )
51, 3, 4mp2an 653 1  |-  E* x E. y ( y  =  A  /\  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   E*wmo 2144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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