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Theorem ssrmof 3218
Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ssrexf.1  |-  F/_ x A
ssrexf.2  |-  F/_ x B
Assertion
Ref Expression
ssrmof  |-  ( A 
C_  B  ->  ( E* x  e.  B  ph 
->  E* x  e.  A  ph ) )

Proof of Theorem ssrmof
StepHypRef Expression
1 ssrexf.1 . . . . 5  |-  F/_ x A
2 ssrexf.2 . . . . 5  |-  F/_ x B
31, 2dfss2f 3146 . . . 4  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
43biimpi 120 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  A  ->  x  e.  B ) )
5 pm3.45 597 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
65alimi 1455 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
7 moim 2090 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph )
)  ->  ( E* x ( x  e.  B  /\  ph )  ->  E* x ( x  e.  A  /\  ph ) ) )
84, 6, 73syl 17 . 2  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  ph )  ->  E* x ( x  e.  A  /\  ph ) ) )
9 df-rmo 2463 . 2  |-  ( E* x  e.  B  ph  <->  E* x ( x  e.  B  /\  ph )
)
10 df-rmo 2463 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
118, 9, 103imtr4g 205 1  |-  ( A 
C_  B  ->  ( E* x  e.  B  ph 
->  E* x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351   E*wmo 2027    e. wcel 2148   F/_wnfc 2306   E*wrmo 2458    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rmo 2463  df-in 3135  df-ss 3142
This theorem is referenced by: (None)
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