Theorem ifpr 2431

Description: Membership of a conditional operator in an unordered pair.

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		ifcl 2384	. . 3 |- ((A e. V /\ B e. V) -> if(ph, A, B) e. V)
2		ifor 2385	. . . 4 |- (if(ph, A, B) = A \/ if(ph, A, B) = B)
3		elprg 2427	. . . 4 |- (if(ph, A, B) e. V -> (if(ph, A, B) e. {A, B} <-> (if(ph, A, B) = A \/ if(ph, A, B) = B)))
4	2, 3	mpbiri 194	. . 3 |- (if(ph, A, B) e. V -> if(ph, A, B) e. {A, B})
5	1, 4	syl 10	. 2 |- ((A e. V /\ B e. V) -> if(ph, A, B) e. {A, B})
6		elisset 1820	. 2 |- (A e. C -> A e. V)
7		elisset 1820	. 2 |- (B e. D -> B e. V)
8	5, 6, 7	syl2an 456	1 |- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  ifcif 2365  {cpr 2414

This theorem is referenced by:  suppr 4599

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-if 2366  df-sn 2416  df-pr 2417

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