Theorem ifbi 3680

Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Assertion

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

Proof of Theorem ifbi

Step	Hyp	Ref	Expression
1		dfbi3 863	. 2 ⊢ ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)))
2		iftrue 3669	. . . 4 ⊢ (φ → if(φ, A, B) = A)
3		iftrue 3669	. . . . 5 ⊢ (ψ → if(ψ, A, B) = A)
4	3	eqcomd 2358	. . . 4 ⊢ (ψ → A = if(ψ, A, B))
5	2, 4	sylan9eq 2405	. . 3 ⊢ ((φ ∧ ψ) → if(φ, A, B) = if(ψ, A, B))
6		iffalse 3670	. . . 4 ⊢ (¬ φ → if(φ, A, B) = B)
7		iffalse 3670	. . . . 5 ⊢ (¬ ψ → if(ψ, A, B) = B)
8	7	eqcomd 2358	. . . 4 ⊢ (¬ ψ → B = if(ψ, A, B))
9	6, 8	sylan9eq 2405	. . 3 ⊢ ((¬ φ ∧ ¬ ψ) → if(φ, A, B) = if(ψ, A, B))
10	5, 9	jaoi 368	. 2 ⊢ (((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) → if(φ, A, B) = if(ψ, A, B))
11	1, 10	sylbi 187	1 ⊢ ((φ ↔ ψ) → if(φ, A, B) = if(ψ, A, B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  ifbid  3681  ifbieq2i  3682