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Theorem ssrmo 23934
Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ssrmo.1  |-  F/_ x A
ssrmo.2  |-  F/_ x B
Assertion
Ref Expression
ssrmo  |-  ( A 
C_  B  ->  ( E* x  e.  B ph  ->  E* x  e.  A ph ) )

Proof of Theorem ssrmo
StepHypRef Expression
1 ssrmo.1 . . . . 5  |-  F/_ x A
2 ssrmo.2 . . . . 5  |-  F/_ x B
31, 2dfss2f 3299 . . . 4  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
43biimpi 187 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  A  ->  x  e.  B ) )
5 pm3.45 808 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
65alimi 1565 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x
( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ph ) ) )
7 moim 2300 . . 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 19 . 2  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  ph )  ->  E* x ( x  e.  A  /\  ph ) ) )
9 df-rmo 2674 . 2  |-  ( E* x  e.  B ph  <->  E* x ( x  e.  B  /\  ph )
)
10 df-rmo 2674 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
118, 9, 103imtr4g 262 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 359   A.wal 1546    e. wcel 1721   E*wmo 2255   F/_wnfc 2527   E*wrmo 2669    C_ wss 3280
This theorem is referenced by:  disjss1f  23969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rmo 2674  df-in 3287  df-ss 3294
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