Theorem nfifd 3355

Description: Deduction version of nfif 3356. (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 3332	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}
2		nfv 1421	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfifd.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2191	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfifd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfand 1460	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
7		nfifd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8	7	nfcrd 2191	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
9	5	nfnd 1547	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
10	8, 9	nfand 1460	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓))
11	6, 10	nford 1459	. . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓)))
12	2, 11	nfabd 2196	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))})
13	1, 12	nfcxfrd 2176	1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629  Ⅎwnf 1349   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-if 3332

This theorem is referenced by:  nfif  3356