Theorem nfiund 51060

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2404. See nfiundg 51061 for a less restrictive version requiring more axioms. (Revised by GG, 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 4956	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfv 1949	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfiund.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfiund.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
5		nfiund.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵)
6	5	nfcrd 2920	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵)
7	3, 4, 6	nfrexdw 3312	. . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
8	2, 7	nfabdw 2947	. 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2925	1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911  ∃wrex 3090  ∪ ciun 4954

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-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-iun 4956

This theorem is used by: (None)