Theorem nfdisjv 4033

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)

Hypotheses

Ref	Expression
nfdisjv.1	⊢ Ⅎ𝑦𝐴
nfdisjv.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfdisjv	⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfdisjv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 4023	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2348	. . . . . 6 ⊢ Ⅎ𝑦𝑥
3		nfdisjv.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4	2, 3	nfel 2357	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfdisjv.2	. . . . . 6 ⊢ Ⅎ𝑦𝐵
6	5	nfcri 2342	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
7	4, 6	nfan 1588	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfmo 2074	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
9	8	nfal 1599	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
10	1, 9	nfxfr 1497	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104  ∀wal 1371  Ⅎwnf 1483  ∃*wmo 2055   ∈ wcel 2176  Ⅎwnfc 2335  Disj wdisj 4021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rmo 2492  df-disj 4022

This theorem is referenced by: (None)