Theorem nfiung 4983

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2401. See nfiun 4981 for a version with more disjoint variable conditions, but not requiring ax-13 2401. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfiung	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiung

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4952	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiung.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiung.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2914	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrex 3360	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfabg 2929	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2920	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907  ∃wrex 3086  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-iun 4952

This theorem is used by: (None)