Theorem reuun1 3515

Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Assertion

Ref	Expression
reuun1	|- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reuun1

Step	Hyp	Ref	Expression
1		ssun1 3392	. 2 |- A C_ ( A u. B )
2		orc 724	. . 3 |- ( ph -> ( ph \/ ps ) )
3	2	rgenw 2605	. 2 |- A. x e. A ( ph -> ( ph \/ ps ) )
4		reuss2 3513	. 2 |- ( ( ( A C_ ( A u. B ) /\ A. x e. A ( ph -> ( ph \/ ps ) ) ) /\ ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) ) -> E! x e. A ph )
5	1, 3, 4	mpanl12 440	1 |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   A.wral 2528   E.wrex 2529   E!wreu 2530   u. cun 3218   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by: (None)