Theorem ifbi 1783

Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-04.)

Assertion

Ref	Expression
ifbi	⊢ ((φ ↔ ψ) → if(φ, A, B) = if(ψ, A, B))

Proof of Theorem ifbi

Step	Hyp	Ref	Expression
1		dfbi 499	. 2 ⊢ ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)))
2		iftrue 1780	. . . 4 ⊢ (φ → if(φ, A, B) = A)
3		iftrue 1780	. . . . 5 ⊢ (ψ → if(ψ, A, B) = A)
4	3	cleqcomd 1106	. . . 4 ⊢ (ψ → A = if(ψ, A, B))
5	2, 4	sylan9eq 1144	. . 3 ⊢ ((φ ∧ ψ) → if(φ, A, B) = if(ψ, A, B))
6		iffalse 1781	. . . 4 ⊢ (¬ φ → if(φ, A, B) = B)
7		iffalse 1781	. . . . 5 ⊢ (¬ ψ → if(ψ, A, B) = B)
8	7	cleqcomd 1106	. . . 4 ⊢ (¬ ψ → B = if(ψ, A, B))
9	6, 8	sylan9eq 1144	. . 3 ⊢ ((¬ φ ∧ ¬ ψ) → if(φ, A, B) = if(ψ, A, B))
10	5, 9	jaoi 275	. 2 ⊢ (((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) → if(φ, A, B) = if(ψ, A, B))
11	1, 10	sylbi 174	1 ⊢ ((φ ↔ ψ) → if(φ, A, B) = if(ψ, A, B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091  ifcif 1776

This theorem is referenced by:  oev 3122  unxpdomlem 3649  ruclem4 4888  ruclem15 4899

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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