Theorem nfdisjv 4113

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 4103	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑦𝑥
3		nfdisjv.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4	2, 3	nfel 2401	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfdisjv.2	. . . . . 6 ⊢ Ⅎ𝑦𝐵
6	5	nfcri 2386	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
7	4, 6	nfan 1618	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfmo 2106	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
9	8	nfal 1629	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
10	1, 9	nfxfr 1527	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104  ∀wal 1400  Ⅎwnf 1513  ∃*wmo 2087   ∈ wcel 2209  Ⅎwnfc 2379  Disj wdisj 4101

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rmo 2536  df-disj 4102

This theorem is referenced by: (None)