Theorem nfdisjv 3993

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 3983	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑦𝑥
3		nfdisjv.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4	2, 3	nfel 2328	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfdisjv.2	. . . . . 6 ⊢ Ⅎ𝑦𝐵
6	5	nfcri 2313	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
7	4, 6	nfan 1565	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfmo 2046	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
9	8	nfal 1576	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
10	1, 9	nfxfr 1474	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104  ∀wal 1351  Ⅎwnf 1460  ∃*wmo 2027   ∈ wcel 2148  Ⅎwnfc 2306  Disj wdisj 3981

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rmo 2463  df-disj 3982

This theorem is referenced by: (None)