Theorem ifpr 2417

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

Assertion

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

Proof of Theorem ifpr

Step	Hyp	Ref	Expression
1		ifcl 2370	. . 3 ⊢ ((A ∈ V ⋀ B ∈ V) → if(φ, A, B) ∈ V)
2		ifor 2371	. . . 4 ⊢ ( if(φ, A, B) = A ⋁ if(φ, A, B) = B)
3		elprg 2413	. . . 4 ⊢ ( if(φ, A, B) ∈ V → ( if(φ, A, B) ∈ {A, B} ↔ ( if(φ, A, B) = A ⋁ if(φ, A, B) = B)))
4	2, 3	mpbiri 194	. . 3 ⊢ ( if(φ, A, B) ∈ V → if(φ, A, B) ∈ {A, B})
5	1, 4	syl 10	. 2 ⊢ ((A ∈ V ⋀ B ∈ V) → if(φ, A, B) ∈ {A, B})
6		elisset 1808	. 2 ⊢ (A ∈ C → A ∈ V)
7		elisset 1808	. 2 ⊢ (B ∈ D → B ∈ V)
8	5, 6, 7	syl2an 454	1 ⊢ ((A ∈ C ⋀ B ∈ D) → if(φ, A, B) ∈ {A, B})

Syntax hints:   → wi 3   ⋁ wo 222   ⋀ wa 223   = wceq 953   ∈ wcel 955  Vcvv 1802   ifcif 2351  {cpr 2400

This theorem is referenced by:  suppr 4562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-if 2352  df-sn 2402  df-pr 2403

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