Theorem iffalse 1781

Description: Value of the conditional operator when its first argument is false.

Assertion

Ref	Expression
iffalse	⊢ (¬ φ → if(φ, A, B) = B)

Proof of Theorem iffalse

Step	Hyp	Ref	Expression
1		dedlemb 570	. . 3 ⊢ (¬ φ → (x ∈ B ↔ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))))
2	1	biabrdv 1184	. 2 ⊢ (¬ φ → B = {x∣((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))})
3		df-if 1777	. 2 ⊢ if(φ, A, B) = {x∣((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
4	2, 3	syl6reqr 1143	1 ⊢ (¬ φ → if(φ, A, B) = B)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ifcif 1776

This theorem is referenced by:  ifbi 1783  elimhyp 1790  elimhyp2v 1791  elimhyp3v 1792  keephyp 1794  keephyp3v 1795  oevn0 3123  unxpdomlem 3649  ruclem13 4897  ruclem20 4904  ruclem21 4905  znnen 4930

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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