Theorem dfif6 4551

Description: An alternate definition of the conditional operator df-if 4549 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfif6	⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfif6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	anbi1d 630	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜑)))
3		eleq1w 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
4	3	anbi1d 630	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝜑)))
5	2, 4	unabw 4326	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜑) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜑))}
6		df-rab 3444	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7		df-rab 3444	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}
8	6, 7	uneq12i 4189	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)})
9		df-if 4549	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜑) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜑))}
10	5, 8, 9	3eqtr4ri 2779	1 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  {cab 2717  {crab 3443   ∪ cun 3974  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-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-un 3981  df-if 4549

This theorem is referenced by:  ifeq1  4552  ifeq2  4553  dfif3  4562