Theorem ifbi 4493

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 1058	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2		iftrue 4476	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		iftrue 4476	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
4	3	eqcomd 2758	. . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵))
5	2, 4	sylan9eq 2807	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6		iffalse 4479	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4479	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2758	. . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵))
9	6, 8	sylan9eq 2807	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
10	5, 9	jaoi 866	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
11	1, 10	sylbi 219	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   = wceq 1550  ifcif 4470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-if 4471

This theorem is referenced by:  ifbid  4494  ifbieq2i  4496  prodeq1i  15918  psdmvr  22203  gsummoncoe1  22340  scmatscm  22542  mulmarep1gsum1  22602  madugsum  22672  mp2pm2mplem4  22838  dchrhash  27301  lgsdi  27364  rpvmasum2  27542  ifnebib  32686  ififcom  32687  itgeq12i  36504  bj-projval  37419  matunitlindflem2  38054  itg2gt0cn  38112  dedths  39524  dfafv2  47664