Theorem ifpr 3774

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

Assertion

Ref	Expression
ifpr	⊢ ((A ∈ C ∧ B ∈ D) → if(φ, A, B) ∈ {A, B})

Proof of Theorem ifpr

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ C → A ∈ V)
2		elex 2867	. 2 ⊢ (B ∈ D → B ∈ V)
3		ifcl 3698	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → if(φ, A, B) ∈ V)
4		ifeqor 3699	. . . 4 ⊢ ( if(φ, A, B) = A ∨ if(φ, A, B) = B)
5		elprg 3750	. . . 4 ⊢ ( if(φ, A, B) ∈ V → ( if(φ, A, B) ∈ {A, B} ↔ ( if(φ, A, B) = A ∨ if(φ, A, B) = B)))
6	4, 5	mpbiri 224	. . 3 ⊢ ( if(φ, A, B) ∈ V → if(φ, A, B) ∈ {A, B})
7	3, 6	syl 15	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → if(φ, A, B) ∈ {A, B})
8	1, 2, 7	syl2an 463	1 ⊢ ((A ∈ C ∧ B ∈ D) → if(φ, A, B) ∈ {A, B})

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ifcif 3662  {cpr 3738

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)