Theorem nfdisjv 3965

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 3955	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2306	. . . . . 6 ⊢ Ⅎ𝑦𝑥
3		nfdisjv.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4	2, 3	nfel 2315	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfdisjv.2	. . . . . 6 ⊢ Ⅎ𝑦𝐵
6	5	nfcri 2300	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
7	4, 6	nfan 1552	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfmo 2033	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
9	8	nfal 1563	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
10	1, 9	nfxfr 1461	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 103  ∀wal 1340  Ⅎwnf 1447  ∃*wmo 2014   ∈ wcel 2135  Ⅎwnfc 2293  Disj wdisj 3953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rmo 2450  df-disj 3954

This theorem is referenced by: (None)