Theorem ifpr 3775

Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)

Assertion

Ref	Expression
ifpr	|- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})

Proof of Theorem ifpr

Step	Hyp	Ref	Expression
1		elex 2868	. 2 |- (A e. C -> A e. _V)
2		elex 2868	. 2 |- (B e. D -> B e. _V)
3		ifcl 3699	. . 3 |- ((A e. _V /\ B e. _V) -> if(ph, A, B) e. _V)
4		ifeqor 3700	. . . 4 |- (if(ph, A, B) = A \/ if(ph, A, B) = B)
5		elprg 3751	. . . 4 |- (if(ph, A, B) e. _V -> (if(ph, A, B) e. {A, B} <-> (if(ph, A, B) = A \/ if(ph, A, B) = B)))
6	4, 5	mpbiri 224	. . 3 |- (if(ph, A, B) e. _V -> if(ph, A, B) e. {A, B})
7	3, 6	syl 15	. 2 |- ((A e. _V /\ B e. _V) -> if(ph, A, B) e. {A, B})
8	1, 2, 7	syl2an 463	1 |- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})

Syntax hints:   -> wi 4   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860  ifcif 3663  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)