Theorem nfifd 3686

Description: Deduction version of nfif 3687. (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 → Ⅎx if(ψ, A, B))

Proof of Theorem nfifd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif2 3665	. 2 ⊢ if(ψ, A, B) = {y ∣ ((y ∈ B → ψ) → (y ∈ A ∧ ψ))}
2		nfv 1619	. . 3 ⊢ Ⅎyφ
3		nfifd.4	. . . . . 6 ⊢ (φ → ℲxB)
4	3	nfcrd 2503	. . . . 5 ⊢ (φ → Ⅎx y ∈ B)
5		nfifd.2	. . . . 5 ⊢ (φ → Ⅎxψ)
6	4, 5	nfimd 1808	. . . 4 ⊢ (φ → Ⅎx(y ∈ B → ψ))
7		nfifd.3	. . . . . 6 ⊢ (φ → ℲxA)
8	7	nfcrd 2503	. . . . 5 ⊢ (φ → Ⅎx y ∈ A)
9	8, 5	nfand 1822	. . . 4 ⊢ (φ → Ⅎx(y ∈ A ∧ ψ))
10	6, 9	nfimd 1808	. . 3 ⊢ (φ → Ⅎx((y ∈ B → ψ) → (y ∈ A ∧ ψ)))
11	2, 10	nfabd 2509	. 2 ⊢ (φ → Ⅎx{y ∣ ((y ∈ B → ψ) → (y ∈ A ∧ ψ))})
12	1, 11	nfcxfrd 2488	1 ⊢ (φ → Ⅎx if(ψ, A, B))

Syntax hints:   → wi 4   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-if 3664

This theorem is referenced by:  nfif  3687