Theorem nfdju 10025

Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)

Hypotheses

Ref	Expression
nfdju.1	⊢ Ⅎ𝑥𝐴
nfdju.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfdju	⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Proof of Theorem nfdju

Step	Hyp	Ref	Expression
1		df-dju 10019	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥{∅}
3		nfdju.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfxp 5691	. . 3 ⊢ Ⅎ𝑥({∅} × 𝐴)
5		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥{1o}
6		nfdju.2	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfxp 5691	. . 3 ⊢ Ⅎ𝑥({1o} × 𝐵)
8	4, 7	nfun 4172	. 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
9	1, 8	nfcxfr 2924	1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

This proof depends on syntax axioms:  Ⅎwnfc 2911   ∪ cun 4152  ∅c0 4281  {csn 4588   × cxp 5656  1_oc1o 8507   ⊔ cdju 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-un 4154  df-opab 5174  df-xp 5664  df-dju 10019

This theorem is used by: (None)