Theorem ifbi 3603

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 467	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
2		notbi 670	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	anbi2d 464	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓)))
4	1, 3	orbi12d 797	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))))
5	4	abbidv 2327	. 2 ⊢ ((𝜑 ↔ 𝜓) → {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))})
6		df-if 3583	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
7		df-if 3583	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))}
8	5, 6, 7	3eqtr4g 2267	1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 712   = wceq 1375   ∈ wcel 2180  {cab 2195  ifcif 3582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-if 3583

This theorem is referenced by:  ifbid  3604  ifbieq2i  3606  ifnebibdc  3628  fodjuomni  7284  fodjumkv  7295  nninfwlpoimlemg  7310  1tonninf  10630  lgsdi  15681  nninfsellemqall  16292  nninfomni  16296