Theorem nfiundg 49775

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2372, see nfiund 49774 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfiundg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4941	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfv 1915	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfiundg.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfiundg.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
5		nfiundg.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵)
6	5	nfcrd 2888	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵)
7	3, 4, 6	nfrexd 3339	. . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
8	2, 7	nfabd 2917	. 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2893	1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  ∃wrex 3056  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-iun 4941

This theorem is referenced by: (None)