Theorem nfifd 4557

Description: Deduction form of nfif 4558. (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		dfif2 4530	. 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))}
2		nfv 1917	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfifd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	3	nfcrd 2892	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
5		nfifd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfimd 1897	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 → 𝜓))
7		nfifd.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
8	7	nfcrd 2892	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
9	8, 5	nfand 1900	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
10	6, 9	nfimd 1897	. . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓)))
11	2, 10	nfabdw 2926	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))})
12	1, 11	nfcxfrd 2902	1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 396  Ⅎwnf 1785   ∈ wcel 2106  {cab 2709  Ⅎwnfc 2883  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-if 4529

This theorem is referenced by:  nfif  4558  nfxnegd  44141