Theorem nfifd 3547

Description: Deduction version of nfif 3548. (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 3521	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}
2		nfv 1516	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfifd.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2322	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfifd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfand 1556	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
7		nfifd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8	7	nfcrd 2322	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
9	5	nfnd 1645	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
10	8, 9	nfand 1556	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓))
11	6, 10	nford 1555	. . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓)))
12	2, 11	nfabd 2328	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))})
13	1, 12	nfcxfrd 2306	1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-if 3521

This theorem is referenced by:  nfif  3548