Theorem ifbi 3348

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anbi2 440	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
2		id 19	. . . . . 6 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
3	2	notbid 592	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	anbi2d 437	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓)))
5	1, 4	orbi12d 707	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))))
6	5	abbidv 2155	. 2 ⊢ ((𝜑 ↔ 𝜓) → {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))})
7		df-if 3332	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
8		df-if 3332	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))}
9	6, 7, 8	3eqtr4g 2097	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  {cab 2026  ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-if 3332

This theorem is referenced by:  ifbid  3349  ifbieq2i  3351