Theorem nfrex 2669

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfrex.1	⊢ ℲxA
nfrex.2	⊢ Ⅎxφ

Assertion

Ref	Expression
nfrex	⊢ Ⅎx∃y ∈ A φ

Proof of Theorem nfrex

Step	Hyp	Ref	Expression
1		dfrex2 2627	. 2 ⊢ (∃y ∈ A φ ↔ ¬ ∀y ∈ A ¬ φ)
2		nfrex.1	. . . 4 ⊢ ℲxA
3		nfrex.2	. . . . 5 ⊢ Ⅎxφ
4	3	nfn 1793	. . . 4 ⊢ Ⅎx ¬ φ
5	2, 4	nfral 2667	. . 3 ⊢ Ⅎx∀y ∈ A ¬ φ
6	5	nfn 1793	. 2 ⊢ Ⅎx ¬ ∀y ∈ A ¬ φ
7	1, 6	nfxfr 1570	1 ⊢ Ⅎx∃y ∈ A φ

Syntax hints:  ¬ wn 3  Ⅎwnf 1544  Ⅎwnfc 2476  ∀wral 2614  ∃wrex 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620

This theorem is referenced by:  r19.12  2727  sbcrexg  3121  nfuni  3897  nfiun  3995  nfop  4604  nfima  4953