Theorem nfifd 3576

Description: Deduction version of nfif 3577. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
nfifd.2	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfifd.3	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfifd.4	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfifd	⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Proof of Theorem nfifd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-if 3550	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}
2		nfv 1539	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfifd.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2346	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfifd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfand 1579	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
7		nfifd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8	7	nfcrd 2346	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
9	5	nfnd 1668	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
10	8, 9	nfand 1579	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓))
11	6, 10	nford 1578	. . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓)))
12	2, 11	nfabd 2352	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))})
13	1, 12	nfcxfrd 2330	1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  Ⅎwnf 1471   ∈ wcel 2160  {cab 2175  Ⅎwnfc 2319  ifcif 3549

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-if 3550

This theorem is referenced by:  nfif  3577