Theorem ifbi 4509

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 1063	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2		iftrue 4492	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		iftrue 4492	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
4	3	eqcomd 2767	. . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵))
5	2, 4	sylan9eq 2816	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6		iffalse 4495	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4495	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2767	. . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵))
9	6, 8	sylan9eq 2816	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
10	5, 9	jaoi 870	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
11	1, 10	sylbi 220	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568  ifcif 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4487

This theorem is referenced by:  ifbid  4510  ifbieq2i  4512  prodeq1i  15969  psdmvr  22311  gsummoncoe1  22447  scmatscm  22649  mulmarep1gsum1  22709  madugsum  22779  mp2pm2mplem4  22945  dchrhash  27411  lgsdi  27474  rpvmasum2  27652  ifnebib  32861  ififcom  32862  itgeq12i  36662  bj-projval  37576  matunitlindflem2  38212  itg2gt0cn  38270  dedths  39682  dfafv2  47814