Theorem nfiund 43536

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem nfiund

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4757	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfv 1957	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfiund.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfiund.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
5		nfiund.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵)
6	5	nfcrd 2927	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵)
7	3, 4, 6	nfrexd 3187	. . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
8	2, 7	nfabd 2954	. 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2933	1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1827   ∈ wcel 2107  {cab 2763  Ⅎwnfc 2919  ∃wrex 3091  ∪ ciun 4755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-iun 4757

This theorem is referenced by: (None)