Theorem ifpr 2485

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 2434	. . 3 ⊢ ((A ∈ V ⋀ B ∈ V) → if(φ, A, B) ∈ V)
2		ifor 2435	. . . 4 ⊢ ( if(φ, A, B) = A ⋁ if(φ, A, B) = B)
3		elprg 2481	. . . 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 192	. . 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 1863	. 2 ⊢ (A ∈ C → A ∈ V)
7		elisset 1863	. 2 ⊢ (B ∈ D → B ∈ V)
8	5, 6, 7	syl2an 456	1 ⊢ ((A ∈ C ⋀ B ∈ D) → if(φ, A, B) ∈ {A, B})

Syntax hints:   → wi 3   ⋁ wo 220   ⋀ wa 221   = wceq 992   ∈ wcel 994  Vcvv 1857   ifcif 2415  {cpr 2468

This theorem is referenced by:  suppr 4733

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-if 2416  df-sn 2470  df-pr 2471

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