Theorem ifbi 4570

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 1050	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2		iftrue 4554	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		iftrue 4554	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
4	3	eqcomd 2746	. . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵))
5	2, 4	sylan9eq 2800	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6		iffalse 4557	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4557	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2746	. . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵))
9	6, 8	sylan9eq 2800	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
10	5, 9	jaoi 856	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
11	1, 10	sylbi 217	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537  ifcif 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-if 4549

This theorem is referenced by:  ifbid  4571  ifbieq2i  4573  prodeq1i  15964  gsummoncoe1  22333  scmatscm  22540  mulmarep1gsum1  22600  madugsum  22670  mp2pm2mplem4  22836  dchrhash  27333  lgsdi  27396  rpvmasum2  27574  ifnebib  32572  itgeq12i  36170  bj-projval  36962  matunitlindflem2  37577  itg2gt0cn  37635  dedths  38918  dfafv2  47047