Theorem ifbi 4513

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 1048	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2		iftrue 4497	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		iftrue 4497	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
4	3	eqcomd 2737	. . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵))
5	2, 4	sylan9eq 2791	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6		iffalse 4500	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4500	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2737	. . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵))
9	6, 8	sylan9eq 2791	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
10	5, 9	jaoi 855	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
11	1, 10	sylbi 216	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ifcif 4491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-if 4492

This theorem is referenced by:  ifbid  4514  ifbieq2i  4516  gsummoncoe1  21712  scmatscm  21899  mulmarep1gsum1  21959  madugsum  22029  mp2pm2mplem4  22195  dchrhash  26656  lgsdi  26719  rpvmasum2  26897  ifnebib  31535  bj-projval  35540  matunitlindflem2  36148  itg2gt0cn  36206  dedths  37497  dfafv2  45484