Theorem nfiund 48105

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2367. See nfiundg 48106 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)

Hypotheses

Ref	Expression
nfiund.1	⊢ Ⅎ𝑥𝜑
nfiund.2	⊢ (𝜑 → Ⅎ𝑦𝐴)
nfiund.3	⊢ (𝜑 → Ⅎ𝑦𝐵)

Assertion

Ref	Expression
nfiund	⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiund

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4998	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfv 1910	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfiund.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfiund.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
5		nfiund.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵)
6	5	nfcrd 2888	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵)
7	3, 4, 6	nfrexdw 3304	. . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
8	2, 7	nfabdw 2923	. 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2898	1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1778   ∈ wcel 2099  {cab 2705  Ⅎwnfc 2879  ∃wrex 3067  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-iun 4998

This theorem is referenced by: (None)