Theorem nfifd 3600

Description: Deduction version of nfif 3601. (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 3574	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}
2		nfv 1552	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfifd.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2363	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfifd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfand 1592	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
7		nfifd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8	7	nfcrd 2363	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
9	5	nfnd 1681	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
10	8, 9	nfand 1592	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓))
11	6, 10	nford 1591	. . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓)))
12	2, 11	nfabd 2369	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))})
13	1, 12	nfcxfrd 2347	1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  Ⅎwnf 1484   ∈ wcel 2177  {cab 2192  Ⅎwnfc 2336  ifcif 3573

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-if 3574

This theorem is referenced by:  nfif  3601