Theorem reuss 2276

Description: Transfer uniqueness to a smaller subclass.

Assertion

Ref	Expression
reuss	|- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)

Distinct variable groups:   x,A   x,B

Proof of Theorem reuss

Step	Hyp	Ref	Expression
1		idd 61	. . . 4 |- (x e. A -> (ph -> ph))
2	1	rgen 1698	. . 3 |- A.x e. A (ph -> ph)
3		reuss2 2275	. . 3 |- (((A (_ B /\ A.x e. A (ph -> ph)) /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)
4	2, 3	mpanl2 707	. 2 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)
5	4	3impb 829	1 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   e. wcel 958  A.wral 1645  E.wrex 1646  E!wreu 1647   (_ wss 2047

This theorem is referenced by:  reuuniss 2889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-reu 1651  df-in 2051  df-ss 2053

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