Theorem ifpr 2425

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 2378	. . 3 ⊢ ((A ∈ V ⋀ B ∈ V) → if(φ, A, B) ∈ V)
2		ifor 2379	. . . 4 ⊢ ( if(φ, A, B) = A ⋁ if(φ, A, B) = B)
3		elprg 2421	. . . 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 1815	. 2 ⊢ (A ∈ C → A ∈ V)
7		elisset 1815	. 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 956   ∈ wcel 958  Vcvv 1809   ifcif 2359  {cpr 2408

This theorem is referenced by:  suppr 4578

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-un 2048  df-if 2360  df-sn 2410  df-pr 2411

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