Theorem nfreud 2783

Description: Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfreud.1	⊢ Ⅎyφ
nfreud.2	⊢ (φ → ℲxA)
nfreud.3	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfreud	⊢ (φ → Ⅎx∃!y ∈ A ψ)

Proof of Theorem nfreud

Step	Hyp	Ref	Expression
1		df-reu 2621	. 2 ⊢ (∃!y ∈ A ψ ↔ ∃!y(y ∈ A ∧ ψ))
2		nfreud.1	. . 3 ⊢ Ⅎyφ
3		nfcvf 2511	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎxy)
4	3	adantl 452	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxy)
5		nfreud.2	. . . . . 6 ⊢ (φ → ℲxA)
6	5	adantr 451	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → ℲxA)
7	4, 6	nfeld 2504	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx y ∈ A)
8		nfreud.3	. . . . 5 ⊢ (φ → Ⅎxψ)
9	8	adantr 451	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
10	7, 9	nfand 1822	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx(y ∈ A ∧ ψ))
11	2, 10	nfeud2 2216	. 2 ⊢ (φ → Ⅎx∃!y(y ∈ A ∧ ψ))
12	1, 11	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃!y ∈ A ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Ⅎwnfc 2476  ∃!wreu 2616

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621

This theorem is referenced by:  nfreu  2785