Theorem nfrexd 2667

Description: Deduction version of nfrex 2670. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfrald.2	⊢ Ⅎyφ
nfrald.3	⊢ (φ → ℲxA)
nfrald.4	⊢ (φ → Ⅎxψ)

Assertion

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

Proof of Theorem nfrexd

Step	Hyp	Ref	Expression
1		dfrex2 2628	. 2 ⊢ (∃y ∈ A ψ ↔ ¬ ∀y ∈ A ¬ ψ)
2		nfrald.2	. . . 4 ⊢ Ⅎyφ
3		nfrald.3	. . . 4 ⊢ (φ → ℲxA)
4		nfrald.4	. . . . 5 ⊢ (φ → Ⅎxψ)
5	4	nfnd 1791	. . . 4 ⊢ (φ → Ⅎx ¬ ψ)
6	2, 3, 5	nfrald 2666	. . 3 ⊢ (φ → Ⅎx∀y ∈ A ¬ ψ)
7	6	nfnd 1791	. 2 ⊢ (φ → Ⅎx ¬ ∀y ∈ A ¬ ψ)
8	1, 7	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃y ∈ A ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  Ⅎwnf 1544  Ⅎwnfc 2477  ∀wral 2615  ∃wrex 2616

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621

This theorem is used by:  nfunid  3899